#6690805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#35927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1861 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,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, 6598001);