We complete measurements of interconversions among the full triad of thermal gradients, charge currents, and spin currents in the topological insulator Bi$_2$Se$_3$ by quantifying the efficiency with which thermal gradients can generate transverse spin currents. We accomplish this by comparing the spin Nernst magneto-thermopower to the spin Hall magnesistance for bilayers of Bi$_2$Se$_3$/CoFeB. We find that Bi$_2$Se$_3$ does generate substantial thermally-driven spin currents. A lower bound for the ratio of spin current to thermal gradient is $J_s/\nabla_x T$ = (4.9 $\pm$ 0.9) $\times$ 10$^{6}$ ($\hbar/2e$) A m$^{-2}$ / K $\mu$m$^{-1}$, and a lower bound for the magnitude of the spin Nernst ratio is $-$0.61 $\pm$ 0.11. The spin Nernst ratio for Bi$_2$Se$_3$ is the largest among all materials measured to date, 2-3 times larger compared to previous measurements for the heavy metals Pt and W.
