Let
X
ν
{X_\nu }
be the set of all permutations
ξ
\xi
of an infinite set
A
A
of cardinality
ℵ
ν
{\aleph _\nu }
with the property: every permutation of
A
A
is a product of two conjugates of
ξ
\xi
. The set
X
0
{X_0}
is shown to be the set of permutations
ξ
\xi
satisfying one of the following three conditions: (1)
ξ
\xi
has at least two infinite orbits. (2)
ξ
\xi
has at least one infinite orbit and infinitely many orbits of a fixed finite size
n
n
. (3)
ξ
\xi
has: no infinite orbit; infinitely many finite orbits of size
k
,
l
k,l
and
k
+
l
k + l
for some positive integers
k
,
l
k,l
; and infinitely many orbits of size
>
2
> 2
. It follows that
ξ
∈
X
0
\xi \in {X_0}
iff some transposition is a product of two conjugates of
ξ
\xi
, and
ξ
\xi
is not a product
σ
i
\sigma i
, where
σ
\sigma
has a finite support and
i
i
is an involution. For
ν
>
0
,
ξ
∈
X
ν
\nu > 0,\;\xi \in {X_\nu }
iff
ξ
\xi
moves
ℵ
ν
{\aleph _\nu }
elements, and satisfies (1), (2) or
(
3
′
)
(3’)
, where
(
3
′
)
(3’)
is obtained from (3) by omitting the requirement that
ξ
\xi
has infinitely many orbits of size
>
2
> 2
. It follows that for
ν
>
0
,
ξ
∈
X
ν
\nu > 0,\;\xi \in {X_\nu }\;
iff
ξ
\xi
moves
ℵ
ν
{\aleph _\nu }
elements and some transposition is the product of two conjugates of
ξ
\xi
. The covering number of a subset
X
X
of a group
G
G
is the smallest power of
X
X
(if any) that equals
G
G
[AH]. These results complete the classification of conjugacy classes in infinite symmetric groups with respect to their covering number.