C₂×#8093479 #72252240

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁶×C₄²

Aliases

direct product

C₂×#8093479

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

54793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 50 points:

(3,4,6,8)(5,7,9,10)(11,12,15,14)(13,16,18,17)(19,20)(21,28)(22,36)(23,38)(24,37)(25,30)(26,27)(29,39)(31,44)(32,34)(33,48)(35,47)(40,45)(41,43)(42,46)(49,50);(5,9)(7,10)(11,13,15,18)(12,16,14,17)(20,27)(21,31)(22,37)(23,39)(25,43)(34,42)(45,48)(47,50);(3,4)(5,7)(6,8)(9,10)(11,14,15,12)(13,16,18,17)(19,21,26,31)(20,28,27,44)(22,32,37,46)(23,40,39,33)(24,42,36,34)(25,35,43,49)(29,48,38,45)(30,47,41,50);(3,5,6,9)(4,7,8,10)(11,13)(12,17)(14,16)(15,18)(19,22)(20,24)(21,32)(23,41)(25,38)(26,37)(27,36)(28,42)(29,43)(30,39)(31,46)(33,47)(34,44)(35,45)(40,50)(48,49);(1,2)(19,23)(20,29)(21,33)(22,30)(24,25)(26,39)(27,38)(28,45)(31,40)(32,50)(34,35)(36,43)(37,41)(42,49)(44,48)(46,47);(19,24)(20,22)(21,34)(23,25)(26,36)(27,37)(28,46)(29,30)(31,42)(32,44)(33,35)(38,41)(39,43)(40,49)(45,47)(48,50);(19,25)(20,30)(21,35)(22,29)(23,24)(26,43)(27,41)(28,47)(31,49)(32,48)(33,34)(36,39)(37,38)(40,42)(44,50)(45,46)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72210038
maximal subgroup of	C₂×#72210036, C₂×#72210038
maximal normal subgroup of	C₂×#72210036, C₂×#72210038

GAP

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

gap> G := Group((3,4,6,8)(5,7,9,10)(11,12,15,14)(13,16,18,17)(19,20)(21,28)(22,36)(23,38)(24,37)(25,30)(26,27)(29,39)(31,44)(32,34)(33,48)(35,47)(40,45)(41,43)(42,46)(49,50), (5,9)(7,10)(11,13,15,18)(12,16,14,17)(20,27)(21,31)(22,37)(23,39)(25,43)(34,42)(45,48)(47,50), (3,4)(5,7)(6,8)(9,10)(11,14,15,12)(13,16,18,17)(19,21,26,31)(20,28,27,44)(22,32,37,46)(23,40,39,33)(24,42,36,34)(25,35,43,49)(29,48,38,45)(30,47,41,50), (3,5,6,9)(4,7,8,10)(11,13)(12,17)(14,16)(15,18)(19,22)(20,24)(21,32)(23,41)(25,38)(26,37)(27,36)(28,42)(29,43)(30,39)(31,46)(33,47)(34,44)(35,45)(40,50)(48,49), (1,2)(19,23)(20,29)(21,33)(22,30)(24,25)(26,39)(27,38)(28,45)(31,40)(32,50)(34,35)(36,43)(37,41)(42,49)(44,48)(46,47), (19,24)(20,22)(21,34)(23,25)(26,36)(27,37)(28,46)(29,30)(31,42)(32,44)(33,35)(38,41)(39,43)(40,49)(45,47)(48,50), (19,25)(20,30)(21,35)(22,29)(23,24)(26,43)(27,41)(28,47)(31,49)(32,48)(33,34)(36,39)(37,38)(40,42)(44,50)(45,46));

C2×#8093479 #72252240