  Eigenstates of permutation gates are either stabilizer states (for gates in
the Pauli group) or magic states, thus allowing universal quantum computation
[M. Planat and Rukhsan-Ul-Haq, Preprint 1701.06443]. We show in this paper that
a subset of such magic states, when acting on the generalized Pauli group,
define (asymmetric) informationally complete POVMs. Such IC-POVMs, investigated
in dimensions $2$ to $12$, exhibit simple finite geometries in their projector
products and, for dimensions $4$ and $8$ and $9$, relate to two-qubit,
three-qubit and two-qutrit contextuality.
