AbstractLet G be a permutation group, acting on a set $$\varOmega $$
Ω
of size n. A subset $${\mathcal {B}}$$
B
of $$\varOmega $$
Ω
is a base for G if the pointwise stabilizer $$G_{({\mathcal {B}})}$$
G
(
B
)
is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of $$\mathrm {Sym}(n)$$
Sym
(
n
)
is large base if there exist integers m and $$r \ge 1$$
r
≥
1
such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd G \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$
Alt
(
m
)
r
⊴
G
≤
Sym
(
m
)
≀
Sym
(
r
)
, where the action of $${{\,\mathrm{Sym}\,}}(m)$$
Sym
(
m
)
is on k-element subsets of $$\{1,\dots ,m\}$$
{
1
,
⋯
,
m
}
and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group $$\mathrm {M}_{24}$$
M
24
in its natural action on 24 points, or $$b(G)\le \lceil \log n\rceil +1$$
b
(
G
)
≤
⌈
log
n
⌉
+
1
. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which $$b(G) > \log n + 1$$
b
(
G
)
>
log
n
+
1
, so our bound is optimal.