Central series quotient of wreath product of groups of order p

Definition

A central series quotient of wreath product of groups of order p is a quotient of a wreath product of groups of order p by a member of its lower central series (which in this case equals the upper central series, because the group is a maximal class group).

Particular cases

1. The abelianization of a wreath product of groups of order $p$ is an elementary abelian group of prime-square order, i.e., a direct product of two copies of the cyclic group of order $p$.
2. The quotient of the group by its commutator with its commutator subgroup is a non-abelian group of order $p^3$. For odd $p$, it is isomorphic to prime-cube order group:U3p, the unique non-abelian $p$-group of order $p^3$ and exponent $p$. For $p = 2$, it is isomorphic to dihedral group:D8.
3. The inner automorphism group of wreath product of groups of order p is a group of order $p^p$. It is a regular p-group. For odd $p$, it is also a maximal class group of exponent $p$, and is not an absolutely regular p-group.

Note that for $p = 2$, (1) and (3) coincide, and (2) coincides with the whole group. For $p = 3$ (2) and (3) coincide.