We calculate the exact amenability constant of the centre of ${{\ell }^{1}}\left( G \right)$ when $G$ is a finite group and is either dihedral, extraspecial, or Frobenius with abelian complement and kernel. This is done using a formula that applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk.

let k be an algebraically closed field of positive characteristic p > 0 and $c \to {\mathbb p}^1_k$ a p-cyclic cover of the projective line ramified in exactly one point. we are interested in the p-sylow subgroups of the full automorphism group autkc. we prove that for curves c with genus 2 or higher, these groups are exactly the extensions of a p-cyclic group by an elementary abelian p-group. the main tool is an efficient algorithm to compute the p-sylow subgroups of autkc starting from an artin–schreier equation for the cover $c \to {\mathbb p}^1_k$. we also characterize curves c with genus $g_c\geq 2$ and a p-group action $g\subset \text{aut}_k c$ such that $2p/(p-1)<|g|/g_c$ and $4/(p-1)^2\leq |g|/g_c^2$. our methods rely on previous work by stichtenoth whose approach we have adopted.