Examples
For example, if G is the fundamental group of a homology sphere, then G is superperfect. The smallest finite, non-trivial superperfect group is the binary icosahedral group (the fundamental group of the Poincaré homology sphere).
The alternating group A5 is perfect but not superperfect: it has a non-trivial central extension, the binary icosahedral group (which is in fact its UCE, and is superperfect). More generally, the projective special linear groups PSL(n, q) are simple (hence perfect) except for PSL(2, 2) and PSL(2, 3), but not superperfect, with the special linear groups SL(n,q) as central extensions. This family includes the binary icosahedral group (thought of as SL(2, 5)) as UCE of A5 (thought of as PSL(2, 5)).
Every acyclic group is superperfect, but the converse is not true: the binary icosahedral group is superperfect, but not acyclic.
Read more about this topic: Superperfect Group
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