Extraspecial group

Remarks

In an extraspecial group, the centre, the commutator subgroup and the Frattini subgroup coincide. The order of an extraspecial group is $p^{1+2n}$ for some $n \ge 1$, and for each $p$ and $n$ there are exactly two extraspecial groups of that order up to isomorphism. Example: the quaternion group $Q_8$ is extraspecial with $p = 2$. Its centre is $\{\pm 1\}$, of order $2$, and $Q_8/\{\pm 1\}$ has order $4$ with every element of order at most $2$, so it is the Klein four-group, which is elementary abelian. Non-example: the cyclic group $\Z/p^2\Z$ is not extraspecial, because its centre is the whole group, of order $p^2$, so clause 1 fails.