Lower bound on size of Schur multiplier for group of prime power order based on minimum size of generating set

Statement
Suppose $$p$$ is a prime number and $$G$$ is a finite p-group with minimum size of generating set $$d$$. Denote by $$G'$$ the derived subgroup of $$G$$ and by $$M(G)$$ the Schur multiplier of $$G$$. Then, $$M(G)$$ is a finite p-group and we have:

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

Related facts

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