Under our assumptions, by [33, Thm. 25.14] the Sylow p-subgroups of HF i are homocyclic abelian, of rank si say.
Products of conjugacy classes in finite and algebraic simple groups
The following lemma shows that these are always nearly homocyclic and they are homocyclic if and only if the offset e is divisible by the dimension d.
Infinite sequences of $p$-groups with fixed coclass