There is one group of genus two
The genus of a finite group \(\Gamma\) is the minimum genus over all closed orientable 2-manifolds in which a Cayley graph for \(\Gamma\) can be imbedded; that is, it is the minimum genus of all Cayley graphs \(G_{\Delta}(\Gamma)\) for \(\Gamma\), the minimum being taken over all possible generating sets \(\Delta\) for \(\Gamma\). There are infinitely many groups of genus zero (the cyclic groups \(Z_ n\), for example) and of genus one \((Z_ n\times Z_ n\), when \(n\geq 3\), for example), but only finitely many groups of a given genus two or more [\textit{T. W. Tucker}, Trans. Am. Math. Soc. 258, 167-179 (1980; Zbl 0444.05039)]. In the present paper, the same author shows that there is exactly one group of genus two: \(\Gamma =<x,y,z| \quad x^ 2=y^ 2=z^ 2=(xy)^ 2=(yz)^ 3=(xz)^ 8=(y(xz)^ 4)^ 2=1>; | \Gamma | =96\).
scientific article; zbMATH DE number 3853100
scientific article