Group of prime power order whose derived subgroup is in the first agemo subgroup of its center

Definition
A group of prime power order whose derived subgroup is in the first agemo subgroup of its center is a group of prime power order $$P$$, i.e., a finite $$p$$-group for some prime number $$p$$, such that the defining ingredient::derived subgroup is contained in the first agemo subgroup of the defining ingredient::center. In symbols:

$$[P,P] \le \mho^1(Z(P))$$

In other words, every commutator is the $$p^{th}$$ power of some central element of the group.