A well-known theorem of Burnside states that any non-linear $\mathbb{C}$-irreducible character of a finite group vanishes on some conjugacy class of the group. My question is then comes naturally:
Are there finite groups for which any non-linear $\mathbb{C}$-irreducible character vanishes on exactly one conjugacy class of the group? Have such groups studied, if exists?
Such groups have been studied by $E.M. Zhmud$ in his paper "On finite groups having an irreducible complex character with one class of zeros(1979)". However, you can very well also refer to this link.