The structures of the title are the even subsets of a six-set and of
an eight-set, viewed modulo set complementation.
The "Brick Space" model of PG(5,2) —
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For the M24 relationship between these spaces, of 15 and of 63 points,
see G. M. Conwell's 1910 paper "The 3-Space PG (3,2) and Its Group,"
as well as Conwell heptads in this journal.