Extraspecial group

Summary

An extraspecial group is a finite $p$-group whose centre has order $p$ and whose quotient by the centre is a nontrivial elementary abelian $p$-group. They are non-abelian $p$-groups that are as close to abelian as possible, and their order is always an odd power of $p$. They occur often inside larger groups, for example in centralisers of involutions in sporadic simple groups.