Inspired by Witten's Wess-Zumino term arguments, I'm curious to know how one calculates homotopy classes more generally for coset spaces. In the above example the coset is $G/H=(SU(3)_L\times SU(3)_R)/SU(3)_{\rm diag}\cong SU(3)$ and so the coset space is itself a group, but how does this extend to more general examples like say $G/H=SU(5)/(SU(3)\times SU(2)\times U(1))$?

What about the case where the groups are non-compact, say they're spacetime symmetry groups? For example, $G/H= ISO(4,1)/ISO(3,1)$ or $G/H=SO(4,2)/SO(3,1)$?