#6690868

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	#36060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6598064);