The edge-cubic spin model on square lattice is studied via Monte Carlo simulation with cluster algorithm. By cooling the system, we found two successive symmetry breakings, i.e., the breakdown of $O_h$ into the group of $C_{3h}$ which then freezes into ground state configuration. To characterize the existing phase transitions, we consider the magnetization and the population number as order parameters. We observe that the magnetization is good at probing the high temperature transition but fails in the analysis of the low temperature transition. In contrast the population number performs well in probing the low- and the high-$T$ transitions. We plot the temperature dependence of the moment and correlation ratios of the order parameters and obtain the high- and low-$T$ transitions at $T_h = 0.602(1)$ and $T_l=0.5422(2)$ respectively, with the corresponding exponents of correlation length $ u_h=1.50(1)$ and $ u_l=0.833(1)$. By using correlation ratio and size dependence of correlation function we estimate the decay exponent for the high-$T$ transition as $eta_h=0.260(1)$. For the low-$T$ transition, $eta_l = 0.267(1)$ is extracted from the finite size scaling of susceptibility. The universality class of the low-$T$ critical point is the same as the 3-state Potts model.
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