The cyclicity rank of empty lattice simplices

We are interested in algebraic properties of empty lattice simplices Δ, that is, d-dimensional lattice polytopes containing exactly d+1 points of the integer lattice ℤ^d. The cyclicity rank of Δ is the minimal number of cyclic subgroups that the quotient group of Δ splits into. It is known that up to dimension d ≤ 4, every empty lattice d-simplex is cyclic, meaning that its cyclicity rank is at most 1. We determine the maximal possible cyclicity rank of an empty lattice d-simplex for dimensions d ≤ 8, and determine the asymptotics of this number up to a logarithmic term.