  For a non-abelian group $G$, its commuting conjugacy class graph
$\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of
conjugacy classes of the non-central elements of $G$ and two distinct vertices
$x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y'
\in y^G$ such that $x'y' = y'x'$. In this paper we compute the genus of
$\mathcal{CCC}(G)$ for six well-known classes of non-abelian two-generated
groups (viz. $D_{2n}, SD_{8n}, Q_{4m}, V_{8n}, U_{(n, m)}$ and $G(p, m, n)$)
and determine whether $\mathcal{CCC}(G)$ for these groups are planar, toroidal,
double-toroidal or triple-toroidal.
