C₂×#8471773 #72331048

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

Aliases

direct product

C₂×#8471773

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

60987 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)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,26);(3,4)(5,9)(6,8)(7,10)(11,13)(12,14)(15,18)(16,17)(20,24)(22,25);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,17)(16,18);(3,4,6,8)(5,9,10,7)(11,14,16,15)(12,17,18,13)(19,21)(20,25)(22,24)(23,26);(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,17)(16,18);(3,5,6,10)(4,7,8,9)(11,15,16,14)(12,13,18,17)(19,22,23,25)(20,21,24,26)

Conjugacy classes & character table

166 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)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,26), (3,4)(5,9)(6,8)(7,10)(11,13)(12,14)(15,18)(16,17)(20,24)(22,25), (3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,17)(16,18), (3,4,6,8)(5,9,10,7)(11,14,16,15)(12,17,18,13)(19,21)(20,25)(22,24)(23,26), (1,2), (3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,17)(16,18), (3,5,6,10)(4,7,8,9)(11,15,16,14)(12,13,18,17)(19,22,23,25)(20,21,24,26));

C2×#8471773 #72331048