C₂×#483592 #72255405

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#483592

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₈	C₂⁴×C₈
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂²×#509	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3039 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 42 points:

(11,12)(13,21)(14,19)(15,20)(16,17)(18,24)(22,34)(23,31)(25,33)(26,28)(27,29)(30,36)(32,38)(35,41)(37,42)(39,40);(3,4,6,8,7,9,10,5)(13,22,24,36)(14,25)(15,27)(18,30,21,34)(19,33)(20,29)(23,35,38,37)(26,39)(28,40)(31,41,32,42);(3,5,10,9,7,8,6,4)(12,17)(13,23,24,38)(14,26)(18,31,21,32)(20,29)(22,35,36,37)(25,39)(30,41,34,42);(11,13,14,22,15,23,25,37,16,24,26,36,27,38,39,35)(12,18,19,30,20,32,33,41,17,21,28,34,29,31,40,42);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216441, C₂×#72216810, C₂×#72216788, …
maximal normal subgroup of	C₂×#72216441, C₂×#72216810, C₂×#72216788, …
maximal quotient of	C₂×#72227052, C₂×#72227144, #72321338, …

GAP

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

gap> G := Group((11,12)(13,21)(14,19)(15,20)(16,17)(18,24)(22,34)(23,31)(25,33)(26,28)(27,29)(30,36)(32,38)(35,41)(37,42)(39,40), (3,4,6,8,7,9,10,5)(13,22,24,36)(14,25)(15,27)(18,30,21,34)(19,33)(20,29)(23,35,38,37)(26,39)(28,40)(31,41,32,42), (3,5,10,9,7,8,6,4)(12,17)(13,23,24,38)(14,26)(18,31,21,32)(20,29)(22,35,36,37)(25,39)(30,41,34,42), (11,13,14,22,15,23,25,37,16,24,26,36,27,38,39,35)(12,18,19,30,20,32,33,41,17,21,28,34,29,31,40,42), (1,2));

C2×#483592 #72255405