Finding Symmetric Groups In The SOMA Cube Puzzle

The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. (https://en.wikipedia.org/wiki/Soma_cube). I understand that the cube puzzle solutions can be found by computer but that doesn't really show the relationship of the solutions to each other which I think could be mathematically and aesthetically interesting. There is the SOMAP which is quite beautiful but it is hard to navigate and has been recently shown to contain non unique solutions (https://www.fam-bundgaard.dk/SOMA/NEWS/N030518.HTM)

I believe a tree graph can contain all of the solutions which also contains branches that can be described by group symmetric operations. Can you find the symmetric group of the highest order?

Here are some hints:

SOMA Cube Solution Methods 1 https://youtu.be/ZWs7A9yNJ3w

SOMA Cube Solution Methods 2 https://youtu.be/72kyEFsUOIU