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?
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? 


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 $G\times G\times G$. 


The smallest example has order 18. It has a normal elementary abelian subgroup $N$ of order 9, and an element $t$ of order 2 such that $txt=x^{1}$ for all $x \in N$. 


I can't prove this, but the Rubik's cube group might work. Update: so this doesn't work. Thanks for the comments. 

