  We investigate multiqubit permutation-symmetric states with maximal entropy
of entanglement. Such states can be viewed as particular spin states, namely
anticoherent spin states. Using the Majorana representation of spin states in
terms of points on the unit sphere, we analyze the consequences of a
point-group symmetry in their arrangement on the quantum properties of the
corresponding state. We focus on the identification of anticoherent states (for
which all reduced density matrices in the symmetric subspace are maximally
mixed) associated with point-group symmetric sets of points. We provide three
different characterizations of anticoherence, and establish a link between
point symmetries, anticoherence and classes of states equivalent through
stochastic local operations with classical communication (SLOCC). We then
investigate in detail the case of small numbers of qubits, and construct
infinite families of anticoherent states with point-group symmetry of their
Majorana points, showing that anticoherent states do exist to arbitrary order.
