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We classify finite primitive permutation groups having a suborbit of length5. As a corollary, we obtain a classification of finite vertex-primitive graphsof valency 5. In the process, we also classify finite almost simple groups thathave a maximal subgroup isomorphic to $\mathrm{Alt}(5)$ or $\mathrm{Sym}(5)$.

Primitive permutation groups with a suborbit of length 5 and vertex-primitive graphs of valency 5