#6483009

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(11,15)(12,16)(13,14)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,36)(30,38)(31,40)(32,39)(35,42)(37,41)(43,45)(44,48)(46,47);(2,5)(6,8)(10,14)(11,12)(17,19,22,32,23,31,21,33)(18,24,26,40,28,39,27,29)(20,30,36,45,37,44,35,46)(25,38,34,47,42,48,41,43);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,21,35,23,37,22,36)(18,25,27,41,28,42,26,34)(19,30,33,46,31,44,32,45)(24,38,29,43,39,48,40,47);(9,12,13,11)(10,16,14,15)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(10,14)(15,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,29,39,40)(25,34,42,41)(30,45,44,46)(38,43,48,47)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,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, 6390205);