#6681787

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,20)(19,27,31,24)(21,32,34,25)(23,26,36,33)(28,30,40,37)(29,38,44,41)(35,46,47,39)(42,43,48,45);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19,22,31)(18,24,20,27)(21,29,34,44)(23,37,36,30)(25,38,32,41)(26,40,33,28)(35,43,47,45)(39,48,46,42);(10,14)(13,16)(17,20,22,18)(19,28,31,40)(21,32,34,25)(23,26,36,33)(24,37,27,30)(29,42,44,48)(35,39,47,46)(38,43,41,45);(1,4,6,5)(2,7,3,8)(17,21)(18,25)(19,29)(20,32)(22,34)(23,35)(24,38)(26,39)(27,41)(28,42)(30,45)(31,44)(33,46)(36,47)(37,43)(40,48);(1,5,6,4)(2,7,3,8)(19,30)(21,34)(23,36)(24,28)(25,32)(26,33)(27,40)(29,43)(31,37)(38,48)(41,42)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6588983);