I am working on a problem in character theory where I try to bound the derived length of a solvable group using information about its characters. In my specific case, it will be extremely helpful for me if I knew that the center was non-trivial. Here is what I know about the group:

- $G$ is solvable.
- the derived subgroup $G'$ is a $p$-group.
- $G''$ is the unique minimal normal subgroup in G.
- $G$ has a faithful irreducible character.

Of course, a lot of information can be deduced from the above. But in particular, I want to know if $G$ has a non-trivial center. This may not be deduced from the above information, but maybe if some additional condition is satisfied?