TITLE: Bounds on number of conjugacy classes in terms of number of elements of a group ?
QUESTION [7 upvotes]: What are bounds on number of conjugacy classes in terms of number of elements of a group ?
(I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and Geoff Robinson.  Below is original text of the question. (Alexander Chervov) ).

It's about the first step to find an upper bound to the order of a finite group with h conjugacy classes (right or left) that depends only on h. (h a natural non nul integer).
I have some doubts about the rigor of my proof that I am sharing with you so that you can help me find a likely error or an omited step.
I have attached the scan of my proof to this post.
Many thanks

REPLY [4 votes]: In the meantime, Barbara Baumeister, Attila Maróti and Hung P. Tong-Viet
have obtained a better lower bound on the number of conjugacy classes of
a group of given order. -- Namely, in 2015 they have proved that for every
$\epsilon > 0$ there exists a $\delta > 0$ such that every group of order
$n \geq 3$ has at least
$$
  \frac{\delta \log_2(n)}{\log_2(\log_2(n))^{3+\epsilon}}
$$
conjugacy classes. -- See here. In this paper, the authors further
cite a conjecture by Edward A. Bertram which asserts that the number of
conjugacy classes of a finite group is bounded below by the logarithm
for base $3$ of its order. This is basically the best one can expect
to be true, as the Mathieu group ${\rm M}_{22}$ has order
$443520 > 3^{11}$ and only $12$ conjugacy classes. The authors also
show that the conjecture holds for groups with trivial solvable radical.