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For a finite group $G$ let $\sigma(G)$ (the "sum" of $G$) be the least numberof proper subgroups of $G$ whose set-theoretical union is equal to $G$, and$\sigma(G)=\infty$ if $G$ is cyclic. We say that a group $G$ is$\sigma$-elementary if for every non-trivial normal subgroup $N$ of $G$ we have$\sigma(G)<\sigma(G/N)$. In this paper we produce the list of all the$\sigma$-elementary groups of sum up to 25. We also show that$\sigma(\Aut(PSL(2,8)))=29$.

Finite Groups that are the union of at most 25 proper subgroups