  We study complexity in a spin system with infinite range interaction, via the
paradigmatic Lipkin-Meshkov-Glick model, in the thermodynamic limit. Exact
expressions for the Nielsen complexity (NC) and the Fubini-Study complexity
(FSC) are derived, that helps us to establish several distinguishing features
compared to complexity in other known spin models. In a time-independent LMG
model, close to phase transition, the NC diverges logarithmically, much like
the entanglement entropy. Remarkably however, in a time dependent scenario,
this divergence is replaced by a finite discontinuity, as we show by using the
Lewis-Riesenfeld theory of time-dependent invariant operators. The FSC of a
variant of the LMG model shows novel behaviour compared to quasi free spin
models. Namely, it diverges logarithmically when the target (or reference)
state is near the separatrix. Numerical analysis indicates that this is due to
the fact that geodesics starting with arbitrary boundary conditions are
"attracted" towards the separatrix and that near this line, a finite change in
the affine parameter of the geodesic produces an infinitesimal change of the
geodesic length. The same divergence is shared by the NC of this model as well.
