The stability, mechanical properties and anisotropic properties of sound velocities of Ga2Mg5, GaMg2, GaMg, O-Ga2Mg, H-Ga2Mg and Ga5Mg2 are investigated systematically by the first-principles calculation. The cohesive energy and formation enthalpy are obtained and used to estimate the stability of the Ga-Mg binary compounds. GaMg compound is the most stable and has the lowest formation enthalpy as -0.162eV/atom of those GaxMgy compounds. The elastic constants of single crystal, hardness, bulk, shear, Young's modulus and Poisson's ratio of the polycrystalline crystal are obtained and used to estimate the mechanical properties. Ga5Mg2 and H-Ga2Mg have the lager bulk, shear and Young’s modulus and corresponding B/G is low. H-Ga2Mg is harder than the other compounds from the results of Poisson’s ratio. The anisotropic mechanical properties are discussed using the anisotropic index, two-dimensional planar projections on different planes of the bulk and Young's modulus. The Young's modulus of H-Ga2Mg shows the strongly anisotropy of mechanical properties and GaMg2 has the weakest anisotropy among all the compounds.