Complexity of Minimal Faithful Permutation Degree for Fitting-free Groups

In this paper, we investigate the complexity of computing the minimal faithful permutation degree for groups without abelian normal subgroups. When our groups are given as quotients of permutation groups, we establish that this problem is in $\textsf{P}$. Furthermore, in the setting of permutation groups, we obtain an upper bound of $\textsf{NC}$ for this problem. This improves upon the work of Das and Thakkar (STOC 2024), who established a Las Vegas polynomial-time algorithm for this class in the setting of permutation groups.