How to compute quotient subgroup efficiently?

Question

Let $G$ be a finite group given by the table representation and a normal subgroup $H$ of $G$ is given. I want to compute $G/H$ that is quotient group.

Model of computation is RAM

For all pair of $a$ and $b$ in $G$ just check $a.b - b.a = 0$ , but it is going to take much time $O(n^2)$. Is there a faster algorithm to solve this problem?