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Classifying finite groups is hard: the exact number of groups of order $2^n$ is known only for $n\leq 10$. The number of 1-dimensional compact Lie groups with $2^n$ components is expected to lie somewhere between the number of groups of order $2^{n+1}$ and the number of groups of order $2^{n+2}$.
The inclusion of the centralizer in $T$ of a rotation through $2\pi/n$ is `close' to showing that $T$ is non-co-Hopfian: isn't this subgroup isomorphic to a (presumably non-split) central extension with kernel $C_n$ and quotient $T$?
There is a space that you can make from the simplicial complex and the groups assigned to its vertices: the polyhedral product. The fundamental group of this space is the graph product of the groups, which only depends on the 1-skeleton of the complex. But only in the case when the complex is flag is this space an Eilenberg-MacLane space. So you can apply this construction to non-flag complexes, it just doesn't give any new groups, which explains why people normally restrict to flag complexes.
I was itching to answer this question myself, but you beat me to it. My article also gives examples of elements (in degrees $6,8,\ldots 2(p-1)$ that are not detected on proper subgroups. Since all the generators are in degrees at most $2p$, there is really only the Bockstein and $P^1$ to worry about when computing the Steenrod algebra action.
