  It is a recent observation that entanglement classification for qubits is
closely related to local $SL(2,\CC)$-invariants including the invariance under
qubit permutations, which has been termed $SL^*$ invariance. In order to single
out the $SL^*$ invariants, we analyze the $SL(2,\CC)$-invariants of four resp.
five qubits and decompose them into irreducible modules for the symmetric group
$S_4$ resp. $S_5$ of qubit permutations. A classifying set of measures of
genuine multipartite entanglement is given by the ideal of the algebra of
$SL^*$-invariants vanishing on arbitrary product states. We find that low
degree homogeneous components of this ideal can be constructed in full by using
the approach introduced in [Phys. Rev. A 72, 012337 (2005)]. Our analysis
highlights an intimate connection between this latter procedure and the
standard methods to create invariants, such as the $\Omega$-process. As the
degrees of invariants increase, the alternative method proves to be
particularly efficient.
