Synthesis, crystal and electronic structure, and physical properties of the Mo$_6$Ga$_{31}$ endohedral cluster superconductor are reported. The compound has two crystallographic modifications, monoclinic and triclinic, in which the same {Mo$_{12}$Ga$_{62}$} building units are perpendicular or codirectional to each other, respectively. Monoclinic and triclinic structures of Mo$_6$Ga$_{31}$ possess qualitatively the same electronic density of states showing a high peak at the Fermi level. Both modifications are inherently present in the bulk specimen. Due to the proximity effect, bulk Mo$_6$Ga$_{31}$ exhibits single superconducting transition at the critical temperature of 8.2 K in zero magnetic field. The upper critical field, which is 7.8 T at zero temperature, shows clear enhancement with respect to the Werthamer-Helfand-Honenberg prediction. Accordingly, heat capacity measurements indicate strong electron-phonon coupling in the superconducting state with the large ratio of $2Delta(0)/(k_BT_c)=4.5$, where 2$Delta(0)$ is the full superconducting gap at zero temperature.
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