A sharp split between Britain's top police officials and rank-and-file officers is developing in the wake of two days of street rioting that injured 112 policemen and uncounted number of civilians and led to 212 arrests.

The argument - in which press, politicians and citizens are taking part - is over the conflict between the rights of free speech and assembly on the one hand, and authority's duty to preserve the public order.

The issue is not how much force the police are allowed to use: Nobody wants the police more heavily armed, the British being somewhat contemptuous of those in the United States or France who use tear gas or bullets to break up street mobs.

Rather, the argument seems to center on whether marches and demonstrations by extremist groups using violence to gain publicity should be permitted, and if so under what circumstances.

"No police officer likes being exposed to violence," said a Scotland Yard superintendent in the measured tones of a veteran. "But they accept that they often have an unpleasant task to perform and they perform it in a lawful, disciplined way."

Four of the superintendent's constables were hit with flying bricks and bottles here Saturday. Some 4,000 police were then protesting a march by the racist National Front from repeated assaults by about 4,500 people mobilized by the Socialist Workers' Party, a fringe Marxist group.

"I don't hold a candle for the extreme rights," the superintendent said, "but they were exercising their right to march and our function is to insure that they can exercise that lawful right. The extreme left was out to block them. We are doing our job in the middle."

"That's specious nonsense," retorted Anthony Judge, an articulate spokesman for rank-and-file policeman and editor of Police Magazine.

"Senior officers have got to stop pussyfooting, stop insisting that their men must be injured in the sacred name of free speech.

"I don't see any chief constables facing those bricks," Judge said bitterly.

There is wide agreement on one point, however. The police, who faced extreme leftists armed with knives and ammonia bottles as well as bricks, have earned praise for their restraint. They stood their ground armed only with three-foot clubs and rarely pulled them out.Typically, the police wrestled their assailants to the ground.

The Scotland Yard superintendent expected nothing less. "A police officer is trained from the word go that he will be exposed to insult, even attack, but he must act within the law. We are not going to wade in to clear the streets," he said.

Judge, the rank-and-filer, scoffed at this. "Police are not supermen," he said. "When they see a colleague going down under bricks, getting kicked on being able to maintain order without or two constables will become violent."

On one point, the two had no quarrel. Here, policemen pride themselves on being able to maintain order without dangerous weapons. In any event, British police believe that if they use stronger arms so will demonstrators. Rather than preserving peace, they think, weapons will escalate disorder.

For the first time in London, however, the police here copied an Ulster technique, using body-length shields of clear plastic to ward off missiles.

"That was a sad day," said the superintendent. I hope we never use the helmets with plastic visors they wear in Belfast."

Even such obvious defensive devices are disliked, Judge explained, because they provoke cries of "police repression."

"Those shields let the leftists claim they were some kind of working-class heroes."

The battle between left extremists and the National Front are becoming a familiar if sporadic part of the British scene. The Front is a 10-year-old outfit with a single program: Save Britain by driving out the 2 million immigrants from the Caribbean and South Asia who were brought here during full employment.

The Front, several of whose leaders were in the tiny British Nazi Party, adds an anti-Semitic flavor to its race cries at closed meetings. It has never elected a member of parliament, but it is putting up more and more candidates and has drawn more than 20 per cent of the vote for local council offices in some working-class districts.

The Socialist Workers' Party is a fringe on the fringe left. Its assault tactics have been condemned by the more "orthodox" Communist and Trostkyist parties here. Its leaders insist that Front "facism" must be crushed by force, that the Front must be driven from the streets.

The Front specializes in marches, carrying Union Jacks and racist placards, in working-class neighborhoods heavily populated with immigrants. It seeks police protection and invites leftists and immigrants to attack both.

Saturday, the Front took its 500 marchers through Lewisham, a South London slum with many Caribbean residents. The Socialist Workers' Party announced in advance that it would block the march, so the Lewisham town council pleaded with Scotland Yard's new police commissioner, David McNee, to use his authority to ban the demonstration.

He refused.

The Birmingham fray on Monday in the Soho district, another black area, posed different problems. A by-election for a vacant seat in Parliament was held there today, and the Front had put up a candidate. Its meeting was held in a hall, not in the streets, and was billed as a legitimate part of the election campaign.

The police threw a cordon around the hall, and a hundred or so Front members heard their man in relative peace. But again, bricks and acid from the Socialist Workers' group took a toll among the constables.

Both extremist groups are delighted with the weekend's work. The Front has aligned itself with the forces of order and gained national publicity for a theme attracting more and more disgruntled workers. It had hoped to draw enough worker votes to give what has been a safe Labor seat to the Conservatives.

The Socialist Workers are equally pleased. Openly bent on revolutionary change, they too would have welcomed a Labor defeat and are also basking in national publicity.

Prime Minister James Callaghan and his home secretary, Merlyn Rees, have been discussing all week how to end the continuing violence. Rees has already said that the test of a free society is its willingness to protect the speech and assembly of those with whom it disagrees.

But Michael Foot, Callaghan's deputy leader, has argued that society has a right to be protected against the spread of race hate, hinting that Front marches should be banned. Foot's nephew Michael is a leading Socials & Workers' Party member.

William Whitelaw, the Tories' deputy leader, has urged the government to take a leaf from its book in Ulster, where both he and Rees have been ministers.

There rival Catholics and Protestants are both allowed to march - but never in each other's ghetos. Even senior police would like to see the government order National Front marches away from immigrant quarters.

Although Britain enjoys a reputation as a lawful, civil humane society, bloody clashes between citizens and police are an old story. Ten years before London's force was created, the Manchester yeomarry smashed a political rally in St. Peter's Field. Drawing their sabers, the yeomarry maimed or slaughtered 600 people gathered to urge manhood suffrage. The Peterloo massacre of 1819 speeded the passage of the electoral reform bill and served as a profound lesson in the perils of an armed police.

More recently, as many as 120 London police have been injured in a single day, resisting mass pickets trying to close a struck film-processing plant in north London.

Every Saturday, police contend with half a dozen riots amont rival fans at big league soccer matches. The arrest of 100 youthful hooligans is normal.

Last Saturday, as Judge observed, the soccer season opened and hooligans had a field day. So many police had been ordered to Lewisham, "they were stretched too thin" for the stadiums.