We study the properties of dark solitons of the nonlinear Schr?dinger equation with (2n+1)-th order nonlinearity. We give the uniform analytical expression for a static dark soliton and find that the width of the static dark soliton decreases with the increase of the nonlinear power index, and its depth remains unchanged. The evolution behavior of the moving gray soliton is studied, and the general expression of the wave function of the moving gray soliton as a function of space and time is given. It is found that if we give the speed of a moving gray soliton, the density and phase shift decrease as the nonlinear power index increases. The energy of the moving gray soliton decreases with the increase of its speed for a given nonlinear power index. Finally, the numerical simulation is given to verify the analytical results.