A transitive action of a non-cyclic group of order 8 on a set of 8 elements

Can a non-cyclic group of order 8 act transitively on a set of 8 elements? From the orbit-stabilizer theorem I see that the transitive group action of a group of order 8 on a set of 8 elements is necessarily free but I do not see how to conclude that the group has to be cyclic.

• Furthermore, a cyclic group does not need to act transitively. Take, for example the cyclic group generated by $(1,2)(3,4,5)$ (which has order 6). – ahulpke Sep 4 '19 at 23:41

Yes, if the order of $$G$$ is $$8$$, the action of $$G$$ on $$G$$ defined by $$L_g(x)=gx$$ is transitive. Every group acts transitively on itself.