#6474211

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6381407);