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We elucidate the correspondence between a particular class of superconformalfield theories in six dimensions and homomorphisms from discrete subgroups of$SU(2)$ into $E_8$, as predicted from string dualities. We show how this matchworks for homomorphisms from the binary icosahedral group $SL(2,5)$ into $E_8$,correcting previous errors in both the mathematics and physics literature. Weuse this correspondence to list the homomorphisms from binary dihedral groups,the binary tetrahedral group, and the binary octahedral group into $E_8$--anovel mathematical result. The partial ordering specified by renormalizationgroup flows suggests an ordering on these homomorphisms similar to the knownordering of nilpotent orbits of a simple Lie algebra dictated by the Hassediagram.

6D SCFTs and the Classification of Homomorphisms $\Gamma_{ADE} \rightarrow E_8$