If you have a finite presentation of a group $G$, then you can easily derive its abelianization $G^{ab}$ from that, by standard techniques: Basically, you create an integer matrix, with one column for each generator, and one row for each relation, noting in entry $a_{ij}$ how often the generator $g_j$ occurred in the relator $r_i$. Then, compute the Smith normal form of this and you can read of the isomorphism type of $G^{ab}$.

If this happens to be non-trivial, you immediately have a proof that your group is non-trivial. Of course, the converse fails.