Largest automorphism groups of perfect algebras

Prove that over the complex numbers the maximal order of the automorphism group of a perfect n-dimensional evolution algebra is $\max\{\prod_l(l(2^l-1))^{m_l}m_l!:\sum_l lm_l=n\}$.

Background

The proposed maximum is attained by direct sums of cyclic perfect indecomposable evolution algebras. The values have been computationally verified through dimension four, but the general interaction between the diagonal automorphism subgroup and the graph-permutation subgroup remains uncontrolled.