C₂×#8131158 #72331143

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

Aliases

direct product

C₂×#8131158

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

58771 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232290
maximal normal subgroup of	C₂×#72232290

GAP

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

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

C2×#8131158 #72331143