C₂×#496127 #72321147

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#496127

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33719	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 38 points:

(1,2);(3,4,5,6)(7,8)(9,13)(10,15)(11,16)(12,14)(17,20)(18,28)(19,27)(21,23)(22,29)(24,25)(26,33)(30,34)(31,35)(32,36)(37,38);(7,9,10,20,11,18,23,34,12,19,22,35,25,33,38,36)(8,13,15,17,16,28,21,30,14,27,29,31,24,26,37,32);(4,6)(7,8)(9,17,19,31)(10,21)(11,24)(12,14)(13,20,27,35)(15,23)(16,25)(18,32,33,30)(22,37)(26,34,28,36)(29,38);(8,14)(9,18,19,33)(10,22)(13,26,27,28)(16,24)(17,30,31,32)(20,36,35,34)(23,38)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72250867
maximal normal subgroup of	C₂×#72250867

GAP

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

gap> G := Group((1,2), (3,4,5,6)(7,8)(9,13)(10,15)(11,16)(12,14)(17,20)(18,28)(19,27)(21,23)(22,29)(24,25)(26,33)(30,34)(31,35)(32,36)(37,38), (7,9,10,20,11,18,23,34,12,19,22,35,25,33,38,36)(8,13,15,17,16,28,21,30,14,27,29,31,24,26,37,32), (4,6)(7,8)(9,17,19,31)(10,21)(11,24)(12,14)(13,20,27,35)(15,23)(16,25)(18,32,33,30)(22,37)(26,34,28,36)(29,38), (8,14)(9,18,19,33)(10,22)(13,26,27,28)(16,24)(17,30,31,32)(20,36,35,34)(23,38));

C2×#496127 #72321147