Location from Foundation by Isaac Asimov
Hidden on Trantor itself — 'at Star's End' meant the other end of the galaxy from Terminus, which is where you started. Among the librarians of the Imperial Library, the mental scientists worked unseen for centuries.
The greatest hiding place in science fiction: Seldon told everyone the Second Foundation was at 'Star's End.' The First Foundation assumed this meant the opposite edge of the galaxy from Terminus. They searched for centuries. They never thought to look at the other end of the line — Trantor, where Seldon started, at the galaxy's center. The Second Foundationers became librarians. They cataloged books and maintained archives while secretly developing mentalic abilities that would have terrified the First Foundation. Their work was the mirror image of Terminus's: where the First Foundation preserved physical technology, the Second preserved the mathematics of psychohistory and developed the mental sciences needed to guide the plan when it went off-track. The revelation of the Second Foundation's location is one of the series' defining moments — the realization that Seldon's wordplay was both a joke and a lesson about assumptions. 'Star's End' meant exactly what it said. The stars end at the center of the galaxy, where Trantor sits. The most powerful organization in the galaxy was hiding in a library.
Outwardly, the Imperial Library — a vast complex of reading rooms, data archives, and scholar's quarters within the Palace grounds. The Second Foundation's actual facilities are distributed through the Library's infrastructure: meeting rooms that appear on no floor plan, shielded chambers behind false walls of bookshelves, communication arrays disguised as data processing equipment. There is no headquarters in the conventional sense — the Second Foundation is a network of individuals embedded in an existing institution, and their 'building' is the Library itself.
Also known as: Star's End, the Second Foundation, the Imperial Library