  The problem of finding the largest finite group with a certain class number
(number of conjugacy classes), $k(G)$, has been investigated by a number of
researchers since the early 1900's and has been solved by computer for $k(G)
\leq 9$. (For the restriction to simple groups for $k(G) \leq 12$.) One has
also tried to find a general upper bound on $|G|$ in terms of $k(G)$. The best
known upper bound in the general case is in the order of magnitude $|G| \leq
k(G)^{2^{k(G)-1}}$. In this paper we consider the restriction of this
longstanding problem to groups of squarefree order. We derive an explicit
formula for the class number $k(G)$ of any group of squarefree order and we
also obtain an estimate $|G| \leq k(G)^3$ in this case. Combining the two
results we get an efficient way to compute the largest squarefree order a group
with a certain class number can have. We also provide an implementation of this
algorithm to compute the maximal squarefree $|G|$ for arbitrary $k(G)$ and
display the results obtained for $k(G) \leq 100$.
