C₂×#648358 #72341781

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

Aliases

direct product

C₂×#648358

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

57649 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)(7,8,9,10)(15,16,19,22)(17,18,20,21);(11,12,13,14)(15,17,19,20)(16,18,22,21);(1,2)(15,18,19,21)(16,20,22,17);(3,4)(5,6)(7,8)(9,10)(15,17,19,20)(16,21,22,18);(8,10)(11,12)(13,14)(15,17)(16,21)(18,22)(19,20);(3,4)(5,6)(7,8)(9,10)(15,16)(17,21)(18,20)(19,22)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72268526, C₂×#72303265, C₂×#72281143
maximal normal subgroup of	C₂×#72268526, C₂×#72303265, C₂×#72281143

GAP

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

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

C2×#648358 #72341781