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.

The Galactic Center of the Milky Way is a supermassive black hole called Sagittarius A*, which has a mass of approximately 4 million times that of our Sun. While the black hole itself does not spin, the material around it certainly does.

The rotational speed of the material orbiting Sagittarius A* depends on its distance from the black hole. At a distance of about 0.01 light-years (0.003 parsecs) from the black hole, stars in the vicinity orbit Sagittarius A* with speeds of around 1,000 kilometers per second (621 miles per second). However, at a distance of about 1 light-year (0.3 parsecs), the orbital speed of stars drops to around 200 kilometers per second (124 miles per second).

It's important to note that these speeds are relative to the black hole itself, as there is no absolute reference frame in space. Additionally, the orbits of the stars around Sagittarius A* are influenced by the gravity of other stars and objects in the vicinity, which can cause their paths to be perturbed and altered over time.

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