It claims that of the 52 proposals made by the advisory group since 2013 and 2020, 31% of them have seen no or very little action.
Its members said said that on almost half of those proposals there was some but insufficient progress, while on less than a quarter (23%) had “strong progress been made or the proposal delivered”.
“The responses also exposed the lack of joined-up thinking and activity across Whitehall which is impeding progress on social mobility,” the report said.
“The commission concluded that a dedicated unit should be set up at the heart of government to coordinate action and ensure its recommendations were delivered.”
Dame Martina Milburn, the SMC's outgoing chair, said: “Social mobility has never been more important. It is the poor and the young who will suffer most from the economic downturn.
“To succeed, action will need to be driven from the heart of government. At present there is no meaningful coordination between departments on the social mobility agenda, and no single force championing social mobility across the government.”
"Committed to levelling up"
Responding to the report, a government spokesperson said: "We are pleased the commission recognises progress in areas such as improving life chances for poorer groups, boosting mental health support for young people, and keeping disadvantaged pupils in education for longer.
"We remain committed to levelling up opportunity across the country, and continue to do all we can to make sure no-one is left behind as a result of coronavirus.
"We also continue to invest significantly in schools and early years, alongside raising wages and increasing work incentives for the lowest paid families."
But Labour’s shadow work and pensions minister, Kate Green, said: “This report exposes a government with no interest in levelling up Britain.
“After a decade of Tory rule, half of all adults from the poorest backgrounds receive no training at all after leaving school, life expectancy is falling for women in the most deprived areas and 600,000 more children are now living in relative poverty than in 2012.
“It's an appalling record of mismanagement and indifference to the life chances working class families and their children.”