The Coxeter group $D_8$ is an index-135 subgroup of $E_8$. One of the consequences of this is that the rectified 8-orthoplex, whose coordinates can be given as all permutations and sign changes of $\{1,1,0,0,0,0,0,0\}$, can join with 134 rotated copies of itself to give a uniform compound with $E_8$ symmetry. In this compound, vertices that would otherwise be distinct coincide by 63 each, meaning that even though the rectifed 8-orthoplex has 112 vertices, the compound only has 112*135/63=240 vertices.
Although the rectified 8-orthoplex also has $BC_8$ symmetry, this disappears in the compound, meaning that trying to make the corresponding uniform compound using 8-cubes, which have all permutations and sign changes of $\{1,1,1,1,1,1,1,1\}$ as possible coordinates, does not work, since it has two vertex orbits.
It seems that any polytope uniform under $BC_8$ but not $D_8$ symmetry can’t form a uniform 135-compound. My question is whether this is this true.
It might be the case that the two orbits end up coinciding, resulting in only a single orbit, and I think it happens with the one of the polytopes which can have all permutations and sign changes of $\{\sqrt2-1,1,1,1,1,1,1,\sqrt2+1\}$ as coordinates.
