  Let $G$ be a finite non-cyclic group. The non-cyclic graph $\Gamma_G$ of $G$
is the graph whose vertex set is $G\setminus Cyc(G)$, two distinct vertices
being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\{a\in
G: \langle a,b\rangle\ \text{is cyclic for each } b\in G\}$. In this paper, we
classify all finite non-cyclic groups $G$ such that $\Gamma_G$ has
(non)orientable genus one.
