# Element structure of extraspecial groups

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## Contents

This article describes the element structure of extraspecial groups. An extraspecial group of order $p^{1 + 2m}$, with $m \ge 1$ and $p$ a prime number, is a non-abelian group $P$ of that order such that $[P,P] = Z(P) = \Phi(P)$ is a cyclic subgroup of order $p$. We can deduce from this that the quotient group is an elementary abelian group of order $p^{2m}$.
For every prime $p$ and every fixed $m$, there are two isomorphism classes of extraspecial groups of order $p^{1+2m}$, known as the extraspecial group of '+' and '-' type respectively.
conjugacy class sizes size 1 ($p$ times), size $p$ ($p^{2m} - 1$ times)
number of conjugacy classes $p^{2m} + p - 1$