let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have $\frac{d(G)\log|G|}{n^2}\rightarrow 0$? If this is true, can you give a complete list of groups with $\frac{d(G)\log |G|}{n^2}\geq \frac{\log 2}{4}$, i.e. groups which are "worse" than $C\_2$?


I believe the answer to the first question is yes, because a transitive group on $n$ letters needs $\mathcal{O}(\frac{n}{\sqrt{\log n}})$ generators, thus if $\frac{d(G)\log|G|}{n^2}$ is large, then $|G|>e^{cn\sqrt{\log n}}$. But then $G$ must involve quite large alternating or symmetric sections, which should imply that $G$ is actually quite easy to generate. I guess that the answer to the second question involves only subgroups of $S\_4$, although I am not too certain about that.



Can a large transitive permutation group need many generators?
Let $G$ be a permutation group of degree n, and let $r>1$ be the minimal block size of $G$. Also, let $s:=n/r$, so that G may be viewed as a subgroup in the wreath product $R\wr S$, where $R\le Sym(r)$ is primitive, and $S=\pi(G)\le Sym(s)$ is transitive ($\pi$ here denotes projection $G\rightarrow Sym(s)$). Then 
$$d(G)\le \frac{a(R)bs}{\sqrt{\log\_{2}{s}}}+d(S) \text{ ($\ast\ast$})$$
where b is an absolute constant, and $a(R)$ denotes the composition length of R. By a result of Pyber and Guralnick, $a(R) \le C\log\_{2}(r)$ for an absolute constant $C$. Also, as you mentioned, $d(S)\le cs/\sqrt{\log\_{2}{s}}$, for an absolute constant c. (For information on the constants $b,c$, and $C$, and on the bound at ($\ast\ast$), see <http://arxiv.org/abs/1504.07506> .)


Now, we also have $\log{|G|}\le \log{|R\wr S|}\le rs\log{r}+s\log{s}$. This means that your first claim follows when $r>\sqrt{\log{s}}$, for example. Thus, it only remains to deal with the case $r\le \sqrt{\log{s}}$.


As you said, the key seems to be reducing the upper bounds when S is either $Alt(s)$ or $Sym(s)$. However, I think that a bound of the form $d(G)\le c'd(R)+2$ would be difficult to prove in this case, and is probably not true. For example, when $R=C\_2$ and $K$ is the intersection of $G$ with the base group of the wreath product, then the number of generators for $K$ as an $S$-group can grow as $bs/\sqrt{\log{s}}$, where b is as above (see Section 3.1 of the paper I mentioned above). 


**EDIT:** @Jan-Christoph Schlage-Puchta The first part of your question, that is, that for a transitive permutation group of degree $n$, $\frac{d(G)\log{|G|}}{n^{2}}$ tends to $0$ as $n$ tends to $\infty$, has now been proved in <http://arxiv.org/abs/1601.02561> . 


