  In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, $\lambda_n$. Our main
result is that for any minimal generating set of transpositions, for
probabilities $\lambda_n=\frac{1+\epsilon_n}{n-1}$ where $n^{-{1/3}+\delta}\le
\epsilon_n<1$ and $\delta>0$, a random induced subgraph has a.s. a unique
largest component of size $\wp(\epsilon_n)\frac{1+\epsilon_n}{n-1}n!$, where
$\wp(\epsilon_n)$ is the survival probability of a specific branching process.
