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  "body": "\n## Definition\n\nLet $p$ be a prime and let $P$ be a finite [[$p$-group|06QYRTX9KQ]].\nLet $Z(P)$ be the [[centre|0DZ93493VN]] of $P$.\nThe group $P$ is *extraspecial* when:\n\n1. $Z(P)$ has order $p$;\n2. the [[quotient group|0KYX2N69J0]] $P/Z(P)$ is a nontrivial [[elementary abelian|DZSZ8ZS554]] $p$-group.\n\n## Remarks\n\nIn an extraspecial group, the centre, the [[commutator subgroup|0ASGSTXSBW]] and the [[Frattini subgroup|KXJG5VV340]] coincide.\nThe 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.\nExample: the [[quaternion group|0NRJGW864D]] $Q_8$ is extraspecial with $p = 2$.\nIts 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.\nNon-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.\n\n## Explanations\n\n### L1 human\n\nAn extraspecial group is a $p$-group with a centre of order $p$ such that the group becomes elementary abelian, and not trivial, once the centre is factored out.\n",
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