I had not realised that Stowe’s table https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=38 would fit snugly inside an octahedron.
If each layer is enclosed inside a square rotated 45 degrees then the top down view looks like the attached image. Eight layers inside an octahedron has a nice congruence. The octahedron, in turn, would fit inside a cube.
The number of elements in each layer is a completely regular at 2-8-18-32-32-18-8-2 = 120. Each plane has two axes of symmetry and there is one vertical axis of symmetry.
I see that the sequence of elements in all four blocks goes clockwise.
René