  We explore nonlocality of three-qubit pure symmetric states shared between
Alice, Bob and Charlie using the Clauser-Horne-Shimony-Holt (CHSH) inequality.
We make use of the elegant parametrization in the canonical form of these
states, proposed by Meill and Meyer (Phys. Rev. A, 96, 062310 (2017)) based on
Majorana geometric representation. The reduced two-qubit states, extracted from
an arbitrary pure entangled symmetric three-qubit state do not violate the CHSH
inequality and hence they are CHSH-local. However, when Alice and Bob perform a
CHSH test, after conditioning over measurement results of Charlie, nonlocality
of the state is revealed. We have also shown that two different families of
three-qubit pure symmetric states, consisting of two and three distinct spinors
(qubits) respectively, can be distinguished based on the strength of violation
in the conditional CHSH nonlocality test. Furthermore, we identify six of the
46 classes of tight Bell inequalities in the three-party, two-setting,
two-outcome i.e., (3,2,2) scenario (Phys. Rev. A 94, 062121 (2016)). Among the
two inequivalent families of three-qubit pure symmetric states, only the states
belonging to three distinct spinor class show maximum violations of these six
tight Bell inequalities.
