We report the synthesis and physical properties of a full Heusler compound, MgPd$_2$Sb, which we found to show superconductivity below $T_c$ = 2.2 K. MgPd$_2$Sb was obtained by a two-step solid-state reaction method and its purity and cubic crystal structure (Fm-3m, a=6.4523(1) r{A}) were confirmed by powder x-ray diffraction. Normal and superconducting states were studied by electrical resistivity, magnetic susceptibility, and heat capacity measurements. The results show that MgPd$_2$Sb is a type-II, weak coupling superconductor ($lambda_{e-p}$ = 0.53). The observed pressure dependence of $T_c$ ($Delta T_c / p approx $ -0.23 K/GPa) is one of the strongest reported for a superconducting Heusler compound. The electronic structure, phonons, and electron-phonon coupling in MgPd$_2$Sb were theoretically investigated. The obtained results are in agreement with the experiment, confirming the electron-phonon coupling mechanism of superconductivity. We compare the superconducting parameters tothose of all reported Heusler-type superconductors.
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