I wonder what the classifying space of the Thompson group or any Sporadic group would be. This has lead me to read two important results that I think would help this list:
1) If a homomorphism $G\to H$ induces an isomorphism on p-localized cohomology $H^*(H)_{(p)}\cong H^*(G)_{(p)}$, then it induces a homotopy equivalence of their (p-localized) classifying spaces.
2) Under p-localization, there is a homotopy equivalence $hocolim_A(EG/C_G(A))\simeq BG$, where A ranges over the elementary abelian p-subgroups of G.
In particular, this helps us get $BM_{12}= BGL_3(\mathbb{F}_3)$ for the Mathieu group, under p-localization for the groups' common Sylow subgroup.
