We explored the magnetic and magnetocaloric properties of the C56 Fullerene-like system formed by the mixed spins 2 and 3/2. We applied Monte Carlo simulations using the Metropolis algorithm and show the effects of different physical parameters. We also established the ground state phase diagrams, at zero reduced temperature, in different planes formed by the reduced external magnetic field, the reduced coupling interaction between the mixed spins, and the reduced crystal field. We found that the total magnetizations predict a specific reduced critical temperature value, confirmed by the peaks observed in the magnetic susceptibilities. To elucidate the behavior of the magnetic entropy changes -ΔSm, of the C56 system, for several external magnetic field values we show that the reduced critical temperature value tC increases when increasing the reduced external magnetic field. In addition, we show the dependency of the maximum “magnetic entropy change” as a function of the reduced external magnetic field. We found an almost linear increase for this parameter when increasing the reduced external magnetic field values. The obtained hysteresis cycles of the studied system showed a decreasing effect of the reduced crystal field followed by a decrease of the magnetic hysteresis surface cycles, as well as the corresponding reduced coercive magnetic field values.