Why didn’t I think of this sooner?!
For the upcoming Anthrocon theme of secret societies, how about . . .
The Triomino Commission.
A cabal of fifty-six elite masters of triangulation. Rumor has it that members identify (and sometimes even address) each other by the three digits* on the corners of a Pressman game tile.
As founder, I claim #000. Who else is in? What is your number?
*Digits range from 0 to 5 and do not descend. (E.g., 223 is valid but 213 is not.)