Bounds on number of conjugacy classes in terms of number of elements of a group ? 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
 A: See MR1182481 (93i:20028) Pyber, L., Finite groups have many conjugacy classes, J. London Math. Soc. (2) 46 (1992), no. 2, 239–249. From the review by I. Ya. Subbotin: 
Let $k(G)$ denote the number of conjugacy classes of a finite group $G$. R. Brauer observed that for every group $G$ of order $n$ we have $k(G)\ge\log\log n$, and proposed the problem of finding substantially better bounds [R. Brauer, in Lectures on modern mathematics, Vol. 1, 133--175, Wiley, New York, 1963; MR0178056 (31 #2314)]. The author proves that every group of order $n$ contains at least $\epsilon\log n/(\log\log n)^8$ conjugacy classes for some fixed $\epsilon$. This essentially settles the problem of Brauer.
A: Zbl 1227.20014. of A. Jaikin-Zapirain paper says that there is conjecture by Ya. Berkovich and V. Zhmud that: number of conj classes > number of prime factors of G (achieved for M_22 and PSL(3,4) )

Quote from Andrei Jaikin-Zapirain paper (Adv. Math. 227, No. 3, 1129-1143 (2011). )

Conjecture. There exists a constant $C > 0$
such that any finite group $G$ of order $n$ satisfies $k(G) \geq  C \log_2 n$.

Main theorem of this paper is the following:

In this paper we establish the first
super-logarithmic lower bound for the
number of conjugacy classes of a finite
nilpotent group.
Theorem 1.1. There exists a
(explicitly computable) constant $C > 0$
such that every finite nilpotent group
$G$ of order $n \geq 8$ satisfies
$$
k(G) > C \frac{\log_2 \log_2 n}{\log_2 \log_2 \log_2 n} \cdot \log_2 n.
$$

Introduction to the paper contains discussion of some history of the subject is quite readable.

On p-groups having the minimal number of conjugacy classes of maximal size.
A. Jaikin-Zapirain, M. F. Newman and E. A. O’Brien

A long-standing question is the
following: do there exist $p$-groups of
odd order having precisely $p-1$
conjugacy classes of the largest
possible size? We exhibit a $3$-group
with this property.


https://doi.org/10.1007/BF01113339
1970 The number of conjugacy classes in a finite group.  Patrick X. Gallagher.
This paper contains results comparing number of conjugacy classes in a group and in its subgroup.

https://arxiv.org/abs/1102.4107
Multiplicities of conjugacy class sizes of finite groups
Hung Ngoc Nguyen

It has been proved recently by Moreto and Craven that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Zaikin-Zapirain. In this note, we prove that if $G$ is a finite simple group then the order of $G$, denoted by $|G|$, is bounded in terms of the largest multiplicity of its conjugacy class sizes and that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group $G$ is $m$, then $|G|$ is bounded in terms of $m$.

A: 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.  
A: There is a theorem of E. Landau which proves that if you fix a positive integer h,
there are only finitely many finite groups with h conjugacy classes. This proof
is more number theory than group theory, in fact. More recently, one person who has
worked more extensively on this question using more group theory is L. Pyber.
