Example of non-abelian groups with these properties

Question

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.

Answer 1

Take, for example, the direct product of a nonabelian group of order $p^3$ with an abelian group of order $p^{a-3}$.

Answer 2

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.