$\DeclareMathOperator\nr{nr}\DeclareMathOperator\rank{rank}$This is a terminology question (I should probably know this, but I don't). Given a group $G$, consider the minimal cardinality $\nr(G)$ of a set $S \subset G$ such that $G$ is the normal closure of $S$: $G = \langle\!\langle S \rangle \!\rangle$ (nr is short for *normal rank*). In other words, how many elements in $G$ do we need to kill to produce the trivial group? What is this invariant called? "Corank" and "normal rank" seem to mean other things, and I'm not sure what other terms to search for.

Also, what methods are there to get a lower bound on $\nr(G)$, say when $G$ is finitely generated? A trivial lower bound in this case is $\rank(H_1(G))$, since clearly $\nr(A)=\rank(A)$ for $A$ a finitely generated abelian group. One has $\nr(G)\leq \rank(G)$, since it suffices to kill a generating set, and if $G\to H$ is a surjection, then $\nr(G)\geq \nr(H)$.