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  "body": "\n## Summary\n\nAn 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.\nThey are non-abelian $p$-groups that are as close to abelian as possible, and their order is always an odd power of $p$.\nThey occur often inside larger groups, for example in centralisers of involutions in sporadic simple groups.\n\n## Explanations\n\n### L1 human\n\nAn extraspecial group is almost abelian.\nAll commutators lie in a tiny central subgroup of order $p$, and the group modulo that subgroup is abelian with every element of order $p$.\nThe quaternion group $Q_8$ and the dihedral group of order $8$ are the two smallest examples, both with $p = 2$.\nFor an odd prime $p$, the group of $3 \\times 3$ upper unitriangular matrices over the field $\\F_p$ is an example of order $p^3$.\n",
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