What are interesting algorithmic questions for groups in table representation? I am currently reading about research problems in nilpotent groups ( assume table representation ). As we know that...

What are interesting algorithmic questions for groups in table representation?
I am currently reading about research problems in nilpotent groups ( assume table representation ). As we know that solvable group isomorphism is known to be in the (almost ) intersection of NP and coNP and even for groups whose derived series has length two, we don't how to do isomorphism in NP ∩ coNP
.
Question 1 : What are the interesting algorithmic questions to answer about nilpotent group class or its sub-class ( which is non-abelian ) other than isomorphism problem?
Question 2 : What are interesting questions to answer about groups when given in the table representation other than isomorphism? A non-interesting question is to find the intersection of two group or to find the order of any element. Testing membership is also not interesting as the input given is a table.
Edit : While class-2 nilpotent groups have long been recognized as the chief bottle- neck in the group isomorphism problem. I am just wondering how much significant it will be if somebody solve isomorphism problem for sub-class of class-2 nilpotent groups ( non-abelian one ) in almost linear time.