The Modular Isomorphism Problem for small groups -- revisiting Eick's algorithm

We study the Modular Isomorphism Problem (MIP) for groups of small order based on an improvement of an algorithm described by B. Eick. Our improvement allows to determine quotients $I(kG)/I(kG)^m$ of the augmentation ideal without first computing the full augmentation ideal $I(kG)$. It allows us to verify that the MIP has a positive answer for groups of order $3^7$ and to significantly reduce the cases that need to be checked for groups of order $5^6$. We further provide a proof for an observation of Bagi\'nski and provide a negative answer to a question of Bleher, Kimmerle, Roggenkamp and Wursthorn.