  The capacity of a quantum gate to produce entangled states on a bipartite
system is quantified in terms of the entangling power. This quantity is defined
as the average of the linear entropy of entanglement of the states produced
after applying a quantum gate over the whole set of separable states. Here we
focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit
space, and calculate the entangling power in terms of the appropriate
local-invariant. A geometric description of the local equivalence classes of
gates is given in terms of the $\mathfrak{su}(3)$ Lie algebra root vectors.
These vectors define a primitive cell with hexagonal symmetry on a plane, and
through the Weyl group the minimum area on the plane containing the whole set
of locally equivalent quantum gates is identified. We give conditions to
determine when a given quantum gate produces maximally entangled states from
separable ones (perfect entanglers). We found that these gates correspond to
one fourth of the whole set of locally-distinct quantum gates. The theory
developed here is applicable to three-level systems in general, where the
non-locality of a quantum gate is related to its capacity to perform non-rigid
transformations on the Majorana constellation of a state. The results are
illustrated by an anisotropic Heisenberg model, the Lipkin-Meshkov-Glick model,
and two coupled quantized oscillators with cross-Kerr interaction.
