The galactic center itself does not have a specific orbital speed as each object that comprises that area of space moves at a unique speed. The easiest answer would include the spin rate of the central black hole, Sagittarius A*, as well as the orbital periods of the closest surrounding stars, the Sagittarius A* cluster.

Per the two linked pages the black hole has an estimated spin rate of 0.1% the speed of light at the event horizon while the stars have orbital velocities between 0.15% and 8% the speed of light.