We show that if $\mathcal S$ is a compact Riemann surface of genus $g=p+1$, where $p$ is prime, with a group of automorphisms $G$ such that $|G|\geq\lambda(g-1)$ for some real number $\lambda>6$, then for all sufficiently large $p$ (depending on $\lambda$), $\mathcal S$ and $G$ lie in one of six infinite sequences of examples. In particular, if $\lambda=8$ then this holds for all $p\geq 17$.