2-Sylow subgroup of symmetric group

Definition

As a Sylow subgroup

The term 2-Sylow subgroup of symmetric group refers to a group that occurs as the 2-Sylow subgroup of a symmetric group on finite set, i.e., a symmetric group on a set of finite size.

For every natural number $n$, there is a corresponding 2-Sylow subgroup of the symmetric group $S_{n}$. For an even number and the odd number bigger than it by 1, the 2-Sylow subgroups of the corresponding symmetric groups are isomorphic.

Explicit description in terms of wreath products

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Particular cases

$n$	symmetric group $S_{n}$	order of $S_{n}$ (= $n!$)	2-Sylow subgroup	order of 2-Sylow subgroup (= largest power of 2 dividing $n!$)	log to the base 2 of order of 2-Sylow subgroup (= sum of the integer parts of $n/2^{k}$, $k$ varying over positive integers)	Information on Sylow subgroup inside symmetric group
0	trivial group	1	trivial group	1	0	--
1	trivial group	1	trivial group	1	0	--
2	cyclic group:Z2	2	cyclic group:Z2	2	1	--
3	symmetric group:S3	6	cyclic group:Z2	2	1	S2 in S3
4	symmetric group:S4	24	dihedral group:D8	8	3	D8 in S4
5	symmetric group:S5	120	dihedral group:D8	8	3	D8 in S5
6	symmetric group:S6	720	direct product of D8 and Z2	16	4
7	symmetric group:S7	5040	direct product of D8 and Z2	16	4
8	symmetric group:S8	40320	wreath product of D8 and Z2	128	7
9	symmetric group:S9	362880	wreath product of D8 and Z2	128	7