The structure of 38 atoms Ag-Cu cluster is studied by using a combination of a genetic algorithm global optimization technique and density functional theory (DFT) calculations. It is demonstrated that the truncated octahedral (TO) Ag32Cu6 core-shell cluster is less stable than the polyicosahedral (pIh) Ag32Cu6 core-shell cluster from the atomistic models and the DFT calculation shows an agreeable result, so the newfound pIh Ag32Cu6 core-shell cluster is further investigated for potential application for O2 dissociation in oxygen reduction reaction (ORR). The activation energy barrier for the O2 dissociation on pIh Ag32Cu6 core-shell cluster is 0.715 eV, where the d-band center is -3.395 eV and the density of states at the Fermi energy level is maximal for the favorable absorption site, indicating that the catalytic activity is attributed to a maximal charge transfer between an oxygen molecule and the pIh Ag32Cu6 core-shell cluster. This work revises the earlier idea that Ag32Cu6 core-shell nanoparticles are not suitable as ORR catalysts and confirms that Ag-Cu nanoalloy is a potential candidate to substitute noble Pt-based catalyst in alkaline fuel cells.