The icosahedral group is the group of symmetries of an icosahedron.
As a symmetry group of one of the Platonic solids, the icosahedral group participates in the ADE pattern:
More in detail, there are variants of the icosahedral group corresponding to the stages of the Whitehead tower of O(3):
the full icosahedral group is the subgroup of O(3)
which is the stabilizer of the standard embedding of the icosahedron into Cartesian space $\mathbb{R}^3$;
the rotational icosahedral group $I \hookrightarrow SO(3)$ is the restriction to orientation-preserving symmetries, hence to SO(3); this is isomorphic to the alternating group $A_5$
finally the binary icosahedral group is the double cover (see at covering of alternating group), hence the lift of $I$ to Spin(3)$\simeq$ SU(2);
next there is a string 2-group lift $\mathcal{I} \hookrightarrow String_{SU(2)}$ of the icosahedral group (Epa 10, Epa-Ganter 16)
Regard the icosahedron, determined uniquely up to isometry on $\mathbb{R}^3$ as a regular convex polyhedron with $20$ faces, as a metric subspace $S$ of $\mathbb{R}^3$. Then the icosahedral group may be defined as the group of isometries of $S$.
More to be added.
The subgroup of orientation-preserving symmetries of the icosahedron is the alternating group $A_5$ whose order is 60. The full icosahedral group is isomorphic to the Cartesian product $A_5 \times \mathbb{Z}_2$ (with the group of order 2).
Hence the order of the icosahedral group is $60 \times 2 = 120$.
There is an exceptional isomorphism $I \simeq PSL_2(\mathbb{F}_5)$, with $2I \simeq SL_2(\mathbb{F}_5)$ covering this isomorphism.
($PSL_2$ the projective special linear group, $SL_2$ the special linear group, $\mathbb{F}_5$ the prime field for $p = 5$)
The coset space $SU(2)/2I$ is the Poincaré homology sphere.
For a little bit about the group cohomology (or at least the homology) of the binary icosahedral group $SL_2(\mathbb{F}_5)$, see Groupprops
Wikipedia, Icosahedral symmetry
Groupprops, Alternating group:A5 Group cohomology of algebrating group:A5
Philip Boalch, The fifty-two icosahedral solutions to Painlevé VI, J. Reine Angew. Math. 596 (2006) 183–214 (arXiv:0406281)
Narthana Epa, Platonic 2-groups, 2010 (pdf)
Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, (arXiv:1605.09192)