C₂×#648004 #72311242

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

Aliases

direct product

C₂×#648004

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

58877 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249699, C₂×#72268519, C₂×#72268526
maximal normal subgroup of	C₂×#72249699, C₂×#72268519, C₂×#72268526
maximal quotient of	#72313741

GAP

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

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

C2×#648004 #72311242