#6830691

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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)(12,14)(13,16)(17,18)(19,30)(20,25)(21,29)(22,27)(23,28)(24,34)(26,39)(31,47)(32,42)(33,45)(35,48)(36,43)(37,46)(38,41)(40,44);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,22)(21,32)(23,34)(25,44)(26,42)(27,46)(28,30)(29,48)(31,33)(35,39)(36,38)(37,40)(41,45)(43,47);(2,5)(6,8)(10,14)(11,16)(17,20,21,33,23,37,39,36)(18,25,29,45,28,46,26,43)(19,31,35,22,34,38,32,40)(24,41,42,44,30,47,48,27);(10,14)(13,15)(17,21,23,39)(18,26,28,29)(19,32,34,35)(20,36,37,33)(22,38,40,31)(24,42,30,48)(25,45,46,43)(27,47,44,41);(1,3,4,7)(2,6,5,8)(17,22,21,38,23,40,39,31)(18,27,29,41,28,44,26,47)(19,33,35,37,34,36,32,20)(24,43,42,25,30,45,48,46)

Polycyclic presentation:

2^9|b^2=fi,c^2=fhi,d^2=i,e^2=gh,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

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, 6737887);