Let $G$ be a finite group, and let $M(G)=H\_2(G,\mathbb{Z})$ be its Schur multiplier. Are there any known bounds on the number of generators of $M(G)$ in terms of $G$? For example, if $G$ is abelian of rank $r$, then $M(G)$ can be generated by $r(r-1)/2$ elements. Is there anything that can be said for more general finite groups?



Number of generators for the Schur multiplier of a finite group
If a finite group has a presentation with $g$ generators and $r$ relations, then the Schur multiplier is generated by $r-g$ elements. There's been lots of work studying groups and presentations where this is minimal ("efficient" presentations and groups).


