If a 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).

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).