In the book on Character Theory by Huppert, the author asserts that
Character table determines nilpotency (class) and solubility (class) of a group.
I have a simple question about this. Everyone agrees that character table of $G$ determines its center and commutator quotient, hence a first step towards nilpotency or solubility.
But then, to determine second center ($Z_2(G)$) or second commutator ($[G',G']$) we have to look for character table of $G/Z(G)$ (or ...). In other words, just character table of $G$ may not be sufficient.
Thus, I was confused with authors above assertion.
Question: Is there any different way to see that character table of a group determines nilpotency or solubility?
To be more simple, I was concerning question like
Question: Suppose we have character table of a $p$-group. (We know that the group is nilpotent even without character table, but can we deduce this from character table? Can we determine its nilpotency class?