Let $a_n$ be the smallest number of conjugacy classes a group with $2^n$ elements can have.
Question 1: What is $a_n$? Can one show that $a_n$ is strict monotone increasing?
The sequence starts for $n=0,1,2,...$ with 1,2,4,5,7,11,13,14.
Question 2: What is $a_{p,n}$ for a fixed prime $p$ more generally, when $a_{p,n}$ is the smallest number of conjugacy classes of a group with $p^n$ elements?
For example for $p=3$ the sequence starts with 1, 3, 9, 11, 17, 19, 41....