K-powerful p-group

Definition for an odd prime p and restricted k
Suppose $$p$$ is an odd prime number, $$k$$ is a nonnegative integer satisfying $$2 \le k \le p - 1$$, and $$P$$ is a p-group. We say that $$P$$ is $$k$$-powerful if $$\gamma_k(P) \le \mho^1(P)$$, where $$\gamma_k(P)$$ denotes the $$k^{th}$$ member of the lower central series of $$P$$ and $$\mho^1(P)$$ is the first agemo subgroup of $$P$$, i.e., the subgroup generated by $$p^{th}$$ powers of elements.