C₂×#8747955 #72300084

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8747955

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

56671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

(3,4)(5,9)(6,7)(8,10)(11,12,14,15)(13,17,18,16)(19,20,22,24)(21,25,26,23);(5,10)(8,9)(19,20)(21,23)(22,24)(25,26);(12,15)(13,18)(19,20)(21,25)(22,24)(23,26);(4,7)(8,9)(11,12)(13,17)(14,15)(16,18)(19,20)(21,25)(22,24)(23,26);(3,4)(5,9)(6,7)(8,10)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,26)(24,25);(3,5)(4,8)(6,10)(7,9)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,26)(24,25);(1,2)

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297795
maximal normal subgroup of	C₂×#72297795

GAP

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

gap> G := Group((3,4)(5,9)(6,7)(8,10)(11,12,14,15)(13,17,18,16)(19,20,22,24)(21,25,26,23), (5,10)(8,9)(19,20)(21,23)(22,24)(25,26), (12,15)(13,18)(19,20)(21,25)(22,24)(23,26), (4,7)(8,9)(11,12)(13,17)(14,15)(16,18)(19,20)(21,25)(22,24)(23,26), (3,4)(5,9)(6,7)(8,10)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,26)(24,25), (3,5)(4,8)(6,10)(7,9)(11,13)(12,16)(14,18)(15,17)(19,21)(20,23)(22,26)(24,25), (1,2));

C2×#8747955 #72300084