Commuting fraction equals five-eighths iff inner automorphism group is Klein four-group

Statement

The following are equivalent for a finite group $G$:

1. The Commuting fraction (?) of $G$ is exactly equal to $5/8$.
2. The Inner automorphism group (?) of $G$ has order equal to four.
3. The Inner automorphism group (?) of $G$ is a Klein four-group.

The condition defines a family of isoclinic groups. The groups in this family that are 2-groups form the $\Gamma _{2}$ family in the Hall-Senior nomenclature.

Related facts

• Commuting fraction more than five-eighths implies abelian