**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 PoincarĂ© homology sphere).

The alternating group *A*_{5} 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 *A*_{5} (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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—Alexander Pope (1688–1744)

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—Bernard Mandeville (1670–1733)