#6680234

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(19,24)(20,25)(21,26)(22,28)(23,27)(29,37)(30,38)(31,43)(32,39)(33,40)(34,41)(35,42)(36,44)(45,47)(46,48);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,32)(25,40)(26,38)(27,39)(28,43)(29,34)(35,45)(36,46)(37,41)(42,47)(44,48);(10,14)(13,16)(17,20,23,35)(18,25,27,42)(19,29,32,46)(21,34,22,36)(24,37,39,48)(26,41,28,44)(30,45,31,33)(38,47,43,40);(2,5)(6,8)(17,21,23,22)(18,26,27,28)(19,30,32,31)(20,34,35,36)(24,38,39,43)(25,41,42,44)(29,45,46,33)(37,47,48,40);(1,3,4,7)(2,6,5,8)(18,27)(19,31)(20,35)(21,22)(24,38)(29,45)(30,32)(33,46)(37,40)(39,43)(41,44)(47,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6587430);