C₂×#7744733 #72302195

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

Aliases

direct product

C₂×#7744733

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

57349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279634
maximal normal subgroup of	C₂×#72279634
maximal quotient of	C₂×#72279950

GAP

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

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

C2×#7744733 #72302195