  The quantum antiferromagnetic Heisenberg model on the fullerene C$_{60}$ in a
magnetic field has $4s$ ground-state magnetization discontinuities with $\Delta
S^z=2$ as a function of the spin quantum number $s$ that disappear at the
classical limit. The molecule can be seen as the fullerene C$_{20}$ with
interpentagon interactions that generate the discontinuities when sufficiently
strong. The discontinuities originate from the antiferromagnetic Ising limit
for both molecules. The results show how spatial symmetry dictates the magnetic
response of the $I_h$ fullerene molecules.
