We present several results connecting the number of conjugacy classes of a finite group on which an irreducible character vanishes, and the size of some centralizer of an element. For example, we show that if
G
G
is a finite group such that
G
≠
G
′
≠
G
G\ne G’\ne G
, then
G
G
has an element
x
x
, such that
|
C
G
(
x
)
|
≤
2
m
|C_G(x)|\le 2m
, where
m
m
is the maximal number of zeros in a row of the character table of
G
G
. Dual results connecting the number of irreducible characters which are zero on a fixed conjugacy class, and the degree of some irreducible character, are included too. For example, the dual of the above result is the following: Let
G
G
be a finite group such that
1
≠
Z
(
G
)
≠
Z
2
(
G
)
1\ne Z(G)\ne Z_2(G)
; then
G
G
has an irreducible character
χ
\chi
such that
|
G
|
χ
2
(
1
)
≤
2
m
\frac {|G|}{\chi ^2(1)}\le 2m
, where
m
m
is the maximal number of zeros in a column of the character table of
G
G
.