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I am looking for examples of non-abelian groups of arbitrarily large size with the following properties

  1. Have order $p^a$, where $a$ is a positive integer and $p$ is prime.
  2. Contain an abelian subgroup of order $p^{a-2}$.

I know one example which is the quaternion group. I am looking for more examples of groups of arbitrarily large size.

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    $\begingroup$ There are too many examples. You need to impose some more restrictions. $\endgroup$ – Derek Holt Nov 26 '18 at 13:05
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Take, for example, the direct product of a nonabelian group of order $p^3$ with an abelian group of order $p^{a-3}$.

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Dihedral groups of order $2^a$ have both properties (the subgroup being the cyclic one generated by the square of a highest-order element), but they also have a larger abelian subgroup of order $2^{a-1}$, so might not be what you're after.

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