Inner automorphism group of wreath product of groups of order p

Definition
Let $$p$$ be a prime number. The inner automorphism group of wreath product of groups of order p is a group obtained as follows: it is the inner automorphism group of the wreath product of groups of order p, where the acting group acts regularly: it acts by cyclic permutation of $$p$$ coordinates.

In other words, if $$\mathbb{Z}_p$$ denotes the cyclic group of order $$p$$, and $$G = \mathbb{Z}_p \wr \mathbb{Z}_p = (\mathbb{Z}_p \times \dots \times \mathbb{Z}_p) \rtimes \mathbb{Z}_p$$, then the group we want is $$H = G/Z(G)$$. It turns out that $$Z(G)$$ is precisely the diagonal subgroup of $$\mathbb{Z}_p \times \dots \times \mathbb{Z}_p$$.

Thus, the group $$H$$ that we want is a group of order $$p^p$$. It turns out that for $$p$$ odd, this group is a maximal class group (for $$p = 2$$, it is isomorphic to the Klein four-group). For all $$p$$, it has exponent p. Since its nilpotence class is $$p - 1$$, it is a regular p-group; however, since it has exponent $$p$$ and size $$p^p$$, it is not an absolutely regular p-group.

Families
Larger collections of groups in which this group figures:


 * Central series quotient of wreath product of groups of order p
 * Inner automorphism group of wreath product of cyclic group of prime power order and group of order p