I need one to test a theory. There are probably many, but I can't seem to think of a single one. My guess is examples are pretty big.


Is there a systematic way to find such examples? Are there databases one can go through to find these things?



What is an example of a finite centerless group with at least 3 generators?
The group $S\_3\wr ({\mathbb Z}\_2 \times {\mathbb Z}\_2 \times {\mathbb Z}\_2)$ where $S\_3$ is the symmetric group with 6 elements, ${\mathbb Z}\_2$ is the group with 2 elements, $\wr$ is the wreath product. 


The fact that it does not have a center is proved by inspection. The fact that it needs at least 3 generators follows from the fact that ${\mathbb Z}\_2^3$ is its quotient. There are lots of similar examples of course. 


 **Update.**  In general , if you take any centerless group $G$ and any group $H$ that needs at least 3 generators, then the wreath product $G\wr H$ has both properties (is centerless and needs at least 3 generators). Another way to construct examples is (as Derek Holt comment below shows) to take any centerless finite group $G$ with nontrivial abelianization and take $G\times G\times G$. 


