#6678177

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2250 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,27,29,23)(20,24,33,41)(22,26,34,42)(28,43,35,37)(30,40,32,38)(31,39,36,44)(45,48,46,47);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(19,28)(20,31)(22,34)(23,37)(24,39)(26,42)(27,43)(29,35)(33,36)(41,44)(45,46)(47,48);(10,14)(13,16)(17,19,21,29)(18,23,25,27)(20,32,33,30)(22,35,34,28)(24,40,41,38)(26,43,42,37)(31,46,36,45)(39,48,44,47);(2,6)(5,7)(19,29)(23,27)(28,35)(30,32)(37,43)(38,40)(45,46)(47,48);(3,8)(5,7)(17,20,21,33)(18,24,25,41)(19,30,29,32)(22,36,34,31)(23,38,27,40)(26,44,42,39)(28,45,35,46)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6585373);