C₂×#493251 #72320454

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#493251

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₈	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4607 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)(9,17)(10,15)(11,16)(12,14)(13,20)(18,24)(19,25)(21,31)(22,28)(23,29)(26,33)(27,35)(30,36)(32,38)(34,37);(7,9,10,18,11,19,21,34,12,20,22,33,23,35,36,32)(8,13,15,26,16,27,31,38,14,17,28,24,29,25,30,37);(8,14)(9,18,20,33)(10,21)(11,23)(13,24,17,26)(15,30)(19,32,35,34)(22,36)(25,37,27,38)(28,31);(8,14)(9,19,20,35)(10,22)(13,25,17,27)(16,29)(18,32,33,34)(21,36)(24,37,26,38)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72250872, C₂×#72250867, #72343493, …
maximal normal subgroup of	C₂×#72250872, C₂×#72250867, #72343493, …

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

C2×#493251 #72320454