C₂×#7812131 #72331883

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

Aliases

direct product

C₂×#7812131

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

At least 9436 conjugacy classes of subgroups; only the 29757 normal subgroups are shown.

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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323407

GAP

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

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

C2×#7812131 #72331883