  We analyze entanglement and nonlocal properties of the convex set of
symmetric $N$-qubits states which are diagonal in the Dicke basis. First, we
demonstrate that within this set, positivity of partial transposition (PPT) is
necessary and sufficient for separability --- which has also been reported
recently in https://doi.org/10.1103/PhysRevA.94.060101 {Phys. Rev. A
\textbf{94}, 060101(R) (2016)}. Further, we show which states among the
entangled DS are nonlocal under two-body Bell inequalities. The diagonal
symmetric convex set contains a simple and extended family of states that
violate the weak Peres conjecture, being PPT with respect to one partition but
violating a Bell inequality in such partition. Our method opens new directions
to address entanglement and non-locality on higher dimensional symmetric
states, where presently very few results are available.
