There are 35 unique heptellations for the 8-simplex, including all permutations of truncations, cantellations, runcinations, sterications, pentellations, and hexications.
The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 8-simplex.
The highest form, the heptihexipentisteriruncicantitruncated 8-simplex is more simply called a omnitruncated 8-simplex with all of the nodes ringed.
A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of: Its 72 vertices represent the root vectors of the simple Lie group A8.
This construction is based on facets of the heptihexipentisteriruncicantitruncated 9-orthoplex, t0,1,2,3,4,5,6,7{37,4} The omnitruncated 8-simplex is the permutohedron of order 9.