The [MO-question][1] asks why the classifying space of a group is not necessarily rationally a product of Eilenberg-Maclane spaces. 
I am looking for classes of examples of connected topological groups/connected associative H-spaces such that their classifying space happens to be rationally a product of Eilenberg-Maclane spaces. I'm the most interested in examples of groups that have an interesting action on a closed manifold.

So far I know that this holds for 

 - compact groups, because the cohomology ring of their classifying space is a polynomial ring
 - Lie groups since they are homotopy equivalent to their maximal compact subgroup
 - the identity component of the homotopy automorphisms of an aspherical space, because its classifying space is already an Eilenberg-Maclane space 

 


  [1]: https://mathoverflow.net/questions/107742/k%C3%BCnneth-formula-for-cohomology