C₂×#686271 #72299474

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#686271

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

59617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

(3,4,5,6)(8,10)(12,14)(16,20)(17,18);(3,4,5,6)(11,12)(13,14)(15,16)(17,22)(18,21)(19,20);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17,19,18)(16,21,20,22);(3,4)(5,6)(15,18,19,17)(16,21,20,22);(3,4)(5,6)(15,16)(17,22)(18,21)(19,20);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297195, C₂×#72298095, C₂×#72298081, …
maximal normal subgroup of	C₂×#72297195, C₂×#72298095, C₂×#72298081, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4,5,6)(8,10)(12,14)(16,20)(17,18), (3,4,5,6)(11,12)(13,14)(15,16)(17,22)(18,21)(19,20), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17,19,18)(16,21,20,22), (3,4)(5,6)(15,18,19,17)(16,21,20,22), (3,4)(5,6)(15,16)(17,22)(18,21)(19,20), (1,2));

C2×#686271 #72299474