A group
H
H
is called capable if it is isomorphic to
G
/
Z
G/\mathbb {Z}
for some group
G
G
. Let
H
H
be a capable group. I. M. Isaacs (2001) showed that if
H
H
is finite, then the index of the centre is bounded above by some function of
|
H
′
|
|H’|
. We show that if
|
H
′
|
>
∞
|H’|>\infty
, then
|
H
:
Z
(
H
)
|
≤
|
H
′
|
c
log
2
|
H
′
|
|H:Z(H)|\leq |H’|^{c\log _2|H’|}
with some constant
c
c
and this bound is essentially best possible. We complete a result of Isaacs, showing that if
H
′
H’
is a cyclic group, then
|
H
:
Z
(
H
)
|
≤
|
H
′
|
2
|H:\mathbf {Z}(H)|\leq |H’|^2
.