  We investigate the intermediate permutational symmetries of a system of
qubits, that lie in between the perfect symmetric and antisymmetric cases. We
prove that, on average, pure states of qubits picked at random with respect to
the uniform measure on the unit sphere of the Hilbert space are almost as
antisymmetric as they are allowed to be. We then observe that multipartite
entanglement, measured by the generalized Meyer- Wallach measure, tends to be
larger in subspaces that are more antisymmetric than the complete symmetric
one. Eventually, we prove that all states contained in the most antisymmetric
subspace are relevant multipartite entangled states in the sense that their
1-qubit reduced states are all maximally mixed.
