Variant of Hopf's formula for Schur multiplier for nilpotent group that uses the free nilpotent group of class one more

Statement

Suppose $G$ is a nilpotent group of nilpotency class $c$. We can calculate the Schur multiplier of $G$ as follows. Let $F$ be a free nilpotent group of class $c + 1$ and $R$ a normal subgroup of $F$ such that $G \cong F/R$. The Schur multiplier $M(G)$ can be computed as: $M(G) \cong (R \cap [F,F])/[F,R]$

This is a variant of Hopf's formula for Schur multiplier. The original version of the formula stipulates that $F$ must be a free group. Note that, when generalizing, what's crucial is to have a class of one more than the class of $G$, because we need a quotient that is big enough to be sensitive to $[F,R]$.

Corresponding formula for exterior square

Suppose $G$ is a nilpotent group of nilpotency class $c$. We can calculate the Schur multiplier of $G$ as follows. Let $F$ be a free nilpotent group of class $c + 1$ and $R$ a normal subgroup of $F$ such that $G \cong F/R$. The exterior square $G \wedge G$ can be computed as: $G \wedge G \cong [F,F]/[F,R]$