# Example of non-abelian groups with these properties

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.

• There are too many examples. You need to impose some more restrictions. – Derek Holt Nov 26 '18 at 13:05

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