Let $n$ be a natural number. For every group $G$ of order $n$, denote

$d(G)$ : The number of elements of the smallest generating set of $G$

How large is the maximum possible value of $d(G)$ depending on $n$ ?

If $n$ is a cyclic number, we have $d(G)=1$ for every group of order $n$. For $n=2p$ , $p$ an odd prime, there are two groups : the cyclic group and the dihedral group with $2$ generators, so in this case the maximum value is $2$.

But I wonder, if the maximal value for $d(G)$ can be determined in general, assuming the factorization of $n$ is known. Is the value known for $n=2048$, for example ?