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In order to enumerate the fake projective planes, as announced in~\cite{CS},we found explicit generators and a presentation for each maximal arithmeticsubgroup $\bar\Gamma$ of~$PU(2,1)$ for which the (appropriately normalized)covolume equals~$1/N$ for some integer~$N\ge1$. Prasad and Yeung \cite{PY1,PY2}had given a list of all such $\bar\Gamma$ (up to equivalence).  The generators were found by a computer search which uses the natural actionof $PU(2,1)$ on the unit ball $B(\C^2)$ in~$\C^2$. Our main results here givecriteria which ensure that the computer search has found sufficiently manyelements of~$\bar\Gamma$ to generate $\bar\Gamma$, and describes a family ofrelations amongst the generating set sufficient to give a presentationof~$\bar\Gamma$.  We give an example illustrating details of how this was done in the case of aparticular~$\bar\Gamma$ (for which $N=864$). While there are no fake projectiveplanes in this case, we exhibit a torsion-free subgroup~$\Pi$ of index~$N$in~$\bar\Gamma$, and give some properties of the surface~$\Pi\backslashB(\C^2)$.

Finding generators and relations for groups acting on the hyperbolic ball