Assoc. Prof. Nick Gill has just uploaded two articles to the preprint arXiv. Both articles are co-authored by Daniele Garzoni, who has just submitted his PhD at the University of Padova. Daniele visited USW in the summer of 2019 and these papers grew out of research undertaken during his visit.
The first paper, entitled "On the number of conjugacy classes of a primitive permutation group with nonabelian socle" describes all of the primitive subgroups of the symmetric group, S_n, that have non-abelian socle and at least n/2 conjugacy classes.
The second paper, entitled "Large minimal invariable generating sets in the finite symmetric groups" uses the bound from the first paper to accurately describe the asymptotics of a statistic, m_I, calculated for the symmetric groups, S_n. The statistic, m_I, is the size of the largest minimal invariable generating set in a group; the paper shows that in S_n, the value of m_I is close to n/2.