The covering number of a group G, denoted by σ (G), is the size of a minimal collection of
proper subgroups of G whose union is G. We investigate which integers are covering
numbers of groups. We determine which integers 129 or smaller are covering numbers, and
we determine precisely or bound the covering number of every primitive monolithic group
with a degree of primitivity at most 129 by introducing effective new computational
techniques. Furthermore, we prove that, if F 1 is the family of finite groups G such that all …

The material presented here is a supplement to the results of the main paper [4]. It contains
calculations of or bounds for the covering number of various monolithic groups with a
degree of primitivity at most 129 that could not be dealt with using the methods of the main
paper such as Algorithms KNS and GKS as well as known results. We use the following
notation in the tables. The notation Mi indicates a conjugacy class of maximal subgroups.
Below the symbol Mi, the number in parentheses indicates the number of conjugate …