Upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order

Statement
Suppose $$p$$ is a prime number and $$G$$ is a finite p-group of order $$p^n$$. Then, the Schur multiplier of $$G$$ is also a finite $$p$$-group and its order is at most $$p^{n(n-1)/2}$$. In fact, we can say something stronger:

$$\! |G'||M(G)| \le p^{n(n-1)/2}$$

where $$G'$$ denotes the derived subgroup of $$G$$.

Related facts

 * Upper bound on size of second cohomology group for groups of prime power order is roughly equivalent in power.

Stronger facts about Schur multiplier

 * Upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order and exponent of center