Given a presentation of a group, together with the promise that the group is finite, is there a computable upper bound on the size of the group? Edited to add: by "presentation" we mean a set of generators and relations. See e.g. Wikipedia.
Define the length of a presentation as the number of generators plus the length of each of the relations in terms of the number of letters. For example, the standard presentation of the cyclic group on $N$ elements has length $N+1$.
It's easy to see that the group can be exponentially larger than its presentation. For example, the cyclic group of size $2^n$ has the following presentation of size about $4n$:
$$ \mathbb Z_{2^n} = \left\langle a_1, \ldots a_n | a_1a_1 = a_2, a_2a_2 = a_3, \ldots a_{n-1}a_{n-1} = a_n, a_na_n = 1\right\rangle. $$
It's plausible to conjecture that exponential is as good as you can do. So a good subquestion is:
Is there a family of finite group presentations for which the size of the group grows super-exponentially in the length of the description of the presentation?