For the finite cyclic group of order $n$, there is the standard presentation $<a|a^n>$. Also for $S_n$ (symmetric group), I know a few presentations where the number of relations is strictly bigger than the number of generators.

I want to get examples of $G$, where $G$ is a finite group with the property that for any presentation of $G$ the number of relations is strictly bigger than the number of generators.

More generally, can one put some condition on a finite group $G$, so that for any presentation the number of relations is strictly bigger than the number of generators?

Thanks in advance!