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The quantum antiferromagnetic Heisenberg model on the fullerene C$_{60}$ in amagnetic field has $4s$ ground-state magnetization discontinuities with $\DeltaS^z=2$ as a function of the spin quantum number $s$ that disappear at theclassical limit. The molecule can be seen as the fullerene C$_{20}$ withinterpentagon interactions that generate the discontinuities when sufficientlystrong. The discontinuities originate from the antiferromagnetic Ising limitfor both molecules. The results show how spatial symmetry dictates the magneticresponse of the $I_h$ fullerene molecules.

$\mathbf{\Delta}$S$^z$=2 quantum magnetization discontinuities proportional in number to the spin s in C$_{60}$: Origin and the role of symmetry