The Nax Empire was an interstellar political entity in the galaxy's Alpha or Beta Quadrants. However, by the time of the Federation, the Empire had long since vanished and was only known through mythological records.
History and specifics[]
At least 150,000 years ago, an all-consuming war between the Nax Empire and the Unified Dray Peoples led to the extinction of both sides.
By the 23rd century, the only known surviving relic of the Empire was Renderer, a warship whose AI continued to operate the vessel in the absence of personnel on the outskirts of the galaxy. (TOS audiobook: A Mirror for Futility)