#6819487

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

1487 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)(11,14)(12,16)(13,15);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,18)(19,28)(20,25)(21,34)(22,38)(23,27)(24,29)(26,36)(30,42)(31,39)(32,41)(33,43)(35,40)(37,48)(44,45)(46,47);(9,12,13,11)(10,14,15,16)(17,19,22,29,23,31,20,32)(18,24,25,28,27,41,38,39)(21,35,36,46,37,30,33,45)(26,42,34,47,43,40,48,44);(2,5)(6,8)(17,20,23,22)(18,25,27,38)(19,29,31,32)(21,33,37,36)(24,39,41,28)(26,34,43,48)(30,45,35,46)(40,47,42,44);(1,3,4,7)(2,6,5,8)(17,21,22,36,23,37,20,33)(18,26,25,34,27,43,38,48)(19,30,29,45,31,35,32,46)(24,40,28,44,41,42,39,47)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=i,e^2=fh,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, 6726683);