  The antiferromagnetic Heisenberg model on the icosahedron, which consists of
20 edge-sharing triangles and belongs to the icosahedral $I_h$ symmetry group,
presents unconventional properties at the classical and quantum level. These
originate in the frustrated nature of the interactions between the spins. For
classical spins the magnetization is discontinuous in a magnetic field. Here we
examine the importance of the connectivity of the icosahedron for the
appearance of the magnetization discontinuity, and also investigate the
transition from the classical to the quantum limit. The influence of
connectivity on the magnetic properties is revealed by considering the cluster
as being made up of a closed strip of a triangular lattice with two additional
spins attached. The classical magnetization discontinuity is shown to evolve
continuously from the discontinuity effected by these two spins when they are
uncoupled to the cluster. In the second part the transition from the classical
to the quantum limit is examined by focusing on the low energy spectrum taking
fully into account the spatial and the spin symmetry of the model in the
characterization of the states. A symmetry analysis of the highly degenerate
due to the connectivity lowest energy classical manifold identifies as its
direct fingerprint the low energy quantum states for spin magnitude as low as
$s=1$, with the latter following a tower of states behavior which relates to
the icosahedron having a structure reminiscent of a depleted triangular
lattice. The classical character of the AHM for small $s$ is also detected on
the ground state energy and correlation functions. On the other hand the
classical magnetization discontinuity in a field eventually disappears for
small $s$, after a weak reentrant behavior.
