How to find out a particular set of permutations cannot generate the whole symmetric group $S_n$

Question

The question is more about the algorithmic side of things. I figure this not-too-inefficient algorithm (if any) in a way is the flip side of the Schreier–Sims algorithm that checks membership of any permutation $\sigma$ w.r.t the $S_n$-subgroup generated by $\{\alpha_1, ..., \alpha_k\}$. Now the question is, given that set of $\alpha$'s, how to find a permutation $\sigma \in S_n$ (if any) that cannot be generated by the set. We can assume $n$ to be sizeable but not too large, while $k$ is small.