I found in Brown's book, Cohomology of groups, that if a group $G$ admits a presentation $\langle g_1, \ldots, g_n \mid r_1, \ldots, r_m \rangle$, then the relation $$\mathrm{rank}~H_2(G) \leq m-n+ \mathrm{rank}~H_1(G)$$ holds. Therefore, if we fix a finite generating set, we are able to find a bound on the number of relations needed just from algebraic properties of the group $G$. My question is:
Do there exist other such relations? For instance, if $n = \mathrm{rank}(G)$, can I find the length of the longest or shortest relation $r_i$ (at least approximately)?