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 $\mathcal{NP}$ and $\text{co}\mathcal{NP}$ and even for groups whose derived series has length two, we don't how to do isomorphism in $\mathcal{NP}$ $\cap$ $\text{co}\mathcal{NP}$. 

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.
First, is it a group at all?  A fundamental problem is checking whether a given operation (in table form) is associative.  The obvious approach is cubic time, Rajagopalan and Schulman do it in near-quadratic time.