  We investigate genuine multipartite nonlocality of pure permutationally
invariant multimode Gaussian states of continuous variable systems, as detected
by the violation of Svetlichny inequality. We identify the phase space settings
leading to the largest violation of the inequality when using displaced parity
measurements, distinguishing our results between the cases of even and odd
total number of modes. We further consider pseudospin measurements and show
that, for three-mode states with asymptotically large squeezing degree,
particular settings of these measurements allow one to approach the maximum
violation of Svetlichny inequality allowed by quantum mechanics. This indicates
that the strongest manifestation of genuine multipartite quantum nonlocality is
in principle verifiable on Gaussian states.
