Genus, thickness and crossing number of graphs encoding the generating properties of finite groups

Assume that $G$ is a finite group and let $a$ and $b$ be non-negative integers. We define an undirected graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if and only $\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$ Our aim is to estimate the genus, the thickness and the crossing number of the graph $\Gamma_{a,b}(G)$ when $a$ and $b$ are positive integers.