#6742416

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,36)(28,43)(29,37)(30,38)(31,39)(32,41)(33,40)(34,44)(35,42)(45,48)(46,47);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19,22,30)(18,23,25,38)(20,31,33,46)(21,28,35,29)(24,39,40,47)(26,43,42,37)(27,34,45,32)(36,44,48,41);(10,14)(13,15)(17,20,22,33)(18,24,25,40)(19,27,30,45)(21,34,35,32)(23,36,38,48)(26,44,42,41)(28,31,29,46)(37,47,43,39);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,36)(28,43)(29,37)(30,38)(31,39)(32,41)(33,40)(34,44)(35,42)(45,48)(46,47);(1,5,6,4)(2,7,3,8)(11,16)(13,15)(18,25)(19,28)(21,35)(23,37)(24,40)(27,31)(29,30)(32,34)(36,47)(38,43)(39,48)(45,46)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6649612);