This is marginally below the 39 per cent of those in positions of power across 37 leading professions, including politicians, journalists, academics, judges and other professions.
The report, Elitist Britain 2019, surveyed 5,000 leading figures across nine broad areas: politics, business, the media, Whitehall and public bodies, public servants, local government, the creative industries, women and sport.
It revealed the 'pipeline' from fee-paying schools through Oxbridge and into top jobs.
Remarkably, the PR consultancy CEO group had the equal highest proportion of privately educated students, equal to entrepreneurs and FTSE 350 chairs, and better than FTSE 350 CEOs (27 per cent) and tech firm CEOs (26 per cent).
Senior judges topped the list of exclusive schooling, with 65 per cent attending fee-paying schools, followed by permanent secretaries (59 per cent), Lords peers (57 per cent), junior ministers (52 per cent) and diplomats (53 per cent).
Of current MPs in the House of Commons in 2017, 29 per cent come from a private school background, four times higher than the electorate they represent.
The five per cent of men’s football internationals who attended independent schools stands in stark contrast to the 37 per cent of rugby internationals and 43 per cent of the England cricket team who had done so.
Nine out of ten PR consultancy chiefs attended university, with a third studying at Russell Group institutions, including seven per cent at Oxbridge.
Similar proportions of FTSE 350 CEOs (36 per cent), Tech firm CEOs (35 per cent) and entrepreneurs (29 per cent) also attended Russell Group universities.
This pales into comparison with the proportion of senior judges (91 per cent), Cabinet Ministers (87 per cent) and MPs (54 per cent) who studied at elite universities.
At the other end of the spectrum, relatively few rugby players (nine per cent) and no cricketers polled went to Russell Group universities.
Sir Peter Lampl, founder and executive chairman of the Sutton Trust, which produced the report with the Social Mobility Commission, said the research shows that Britain is an increasingly divided society.
"Divided by politics, by class, by geography. Social mobility, the potential for those to achieve success regardless of their background, remains low," he said.
"As our report shows, the most influential people across sport, politics, the media, film and TV, are five times as likely to have attended a fee-paying school."
Social Mobility Commission Dame Martina Milburn added: "Politicians, employers and educators all need to work together to ensure that Britain’s elite becomes more diverse in gender, ethnicity and social background. It is time to close the power gap and ensure that those at the top can relate to and represent ordinary people."