The Galactic Mining Company was a mining company that operated the Delta-Vega Station, an unmanned lithium cracking station on the remote planet Delta Vega in 2265. (TOS: "Where No Man Has Gone Before")
The name comes from a sign on the exterior wall of the facility shown in the episode. "Galactic Mining" is clearly visible in the close up, and the long shot, while not clear, suggests that it is followed by CO or Co (a common abbreviation for "Company").