The Talos star group was a stellar cluster in the Milky Way Galaxy which was named after the Talos system. This system was located in the third quadrant of the Vernal Galaxy. In 2254, this star group was approximately two weeks away from the Rigel system at the maximum cruising speed of a Constitution-class heavy cruiser. In 2267, this star group was six days away from Starbase 11 at the maximum warp speed of a Constitution-class heavy cruiser.
By the mid-23rd century, the cluster was unexplored and was not the site of any colonies. The only expedition into the region, a survey by the SS Columbia, had culminated in the Columbia disappearing under mysterious circumstances in 2236. Eighteen years later, in 2254, the USS Enterprise, under the command of Captain Christopher Pike, on a search-and-recovery mission for survivors of the downed survey ship became the second ship to enter the star group. (TOS: "The Cage", "The Menagerie, Part I")
In 2293, the location of Talos in the Milky Way Galaxy was labeled in a star chart that was in Captain James T. Kirk's quarters aboard the USS Enterprise-A. (Star Trek VI: The Undiscovered Country, okudagram)
In the script of "The Cage", the Talos star group was described as "a dual-sun system."
The image of the Talos star group was a color astrophotographic exposure of M45, the Pleiades Cluster. The exposure was relatively short, and so was lacking detail of the characteristic Merope Nebula. However, the nebula does give a blue halo to the stars visible within the photograph.
The vernal galaxy information comes from the "For eyes of Starfleet Command only" file read by Kirk in "The Menagerie, Part I". It was unknown what the vernal galaxy was, exactly, and how its quadrants are arranged. This reference system was never used again. Dialogue in "The Cage" clearly states Earth and Talos to be in the same galaxy. According to Pike, the Enterprise originated from a stellar group at the other end of the galaxy relative to Talos.
According to Star Trek: Star Charts ("United Federation of Planets III") and Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), the Talos Star Group was located in the Alpha Quadrant.