Class-preserving automorphism group of finite p-group is p-group

Statement

Suppose $P$ is a finite $p$-group, i.e., a Group of prime power order (?) where the prime is $p$. Let $\operatorname {Class} (P)$ denote the group of Class-preserving automorphism (?)s of $P$, i.e., the automorphisms that send every element to within its conjugacy class. Then, $\operatorname {Class} (P)$ is also a $p$-group.

Facts used

1. Class-preserving implies stability automorphism of central series
2. Stability group of subnormal series of p-group is p-group

Proof

The proof follows directly from facts (1) and (2).