The calculations of cohomology of homogeneous space $X=G/H$ is reduced to a problem in linear algebra. 

[If $G$ is compact and connected then any form is cohomologous to its left shifts and therefore it is cohomologous to the avagage of all left shifts, which is a left-invariant form.
Thus $H^k(X,\mathbb R)$ is isomorphic to space of $k$-forms at one point which is invariant under roation of the stabilizer.]