I am looking for a list of classifying spaces of groups (discrete and/or topological) along with associated covers; there does not seem to be such cataloging on the web. Or if not a list, just some further fundamental examples. For instance, here are the ones I have off the top of my head:

$B\mathbb{Z}_2=\mathbb{R}P^\infty$ with cover $S^\infty$

$B\mathbb{Z}_n=L_n^\infty$ with cover $S^\infty$

$B\mathbb{Z}=S^1$ with cover $\mathbb{R}$

$BS^1=\mathbb{C}P^\infty$ with cover $S^\infty$

$B(F_2)=S^1\vee S^1$ with cover $T$ (infinite fractal tree)

$BU(n)=G_n(\mathbb{C}^\infty)$ with cover $V_n(\mathbb{C}^\infty)$ (space of orthonormal families of $n$-vectors in $\mathbb{C}^\infty$)

And of course, $B(G_1\times G_2)=BG_1\times BG_2$, so I do not care that much about ''decomposable'' groups.