C₂×#8689280 #72331933

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

Aliases

direct product

C₂×#8689280

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

66575 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:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323412

GAP

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

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

C2×#8689280 #72331933