The wide-ranging survey also discloses that the average-sized primary class in some schools in major cities such as Birmingham, Liverpool and Nottingham is 35 and above. Heads told The TES that, on average, teachers had one more child in their class than the Government has claimed.
The survey - based on a 5 per cent sample of schools - also shows that almost half of all primaries have classes of at least 30. Heads of primary schools in Manchester, Liverpool, Bolton, Barnsley and Chippenham in Wiltshire reported classes of more than 40.
The findings, if extrapolated across the country, would mean that 1.9 million children - or 49 per cent of primary pupils - are taught in classes of 30 and over.
But class size is only one of the pressing problems facing schools, according to primary and secondary heads. Six out of 10 reported deteriorating pupil behaviour and four out of five said the role of governors should be more clearly defined. This issue has been highlighted this week with disputes in schools in Nottingham and Halifax.
The survey's findings should alarm the Government which is under pressure from parents to act on rising classes and from heads who want the power to deal effectively with disruptive pupils.
Class size is almost certain to become an election issue with Labour committed to infant classes of no more than 30.
This week an extra 100,000 pupils started school exacerbating heads' fears that they would be forced to squeeze in even more children following the Government's abolition of statutory minimum space requirements.
Earlier this year ministers admitted that primary pupil-teacher ratios were worse than they were a decade ago. They claimed that the average size of class taught by one teacher was 27.3, but, according to the surveyed heads, it is 28.2.
The survey also highlighted chronic shortages of good quality teachers with primary heads in the North, the South-west and the Midlands reporting serious recruitment problems. Across the country more than a third of primaries said they were unable to find staff of the calibre they needed to teach the national curriculum.
In the secondary sector 53 per cent of heads reported problems with the greatest disquiet in the North-west and South-east.
Four out of five primaries and seven out of 10 secondaries which had problems recruiting staff could not find suitably qualified maths teachers. Almost half the primaries and nearly a third of the secondaries with staffing difficulties were unable to get hold of good English teachers. Secondary schools also had trouble recruiting staff to teach languages and science.
Those findings will provide ammunition for Gillian Shephard, the Education and Employment Secretary, in her fight to secure more cash for schools. She will be heartened to hear that, despite everything, four out of five headteachers claimed morale was good.
The survey of 1,210 schools - 5 per cent of schools in England and Wales - was carried out at the end of the summer term and had a 44 per cent response rate.