#6692777

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gi,d^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,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, 6599973);