#6692779

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(19,28)(24,38)(29,39)(30,37)(31,43)(40,47)(44,48)(45,46);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,29)(20,25)(21,33)(22,27)(23,36)(24,30)(26,32)(28,39)(31,40)(34,42)(35,41)(37,38)(43,47)(44,46)(45,48);(9,12,13,11)(10,14,15,16)(17,19)(18,24)(20,30)(21,31)(22,28)(23,37)(25,39)(26,40)(27,38)(29,36)(32,45)(33,44)(34,46)(35,43)(41,48)(42,47);(2,6)(5,7)(17,20,22,23)(18,25,27,36)(19,30,28,37)(21,34,35,32)(24,39,38,29)(26,33,42,41)(31,46,43,45)(40,44,47,48);(3,8)(5,7)(17,21,22,35)(18,26,27,42)(19,31,28,43)(20,32,23,34)(24,40,38,47)(25,41,36,33)(29,44,39,48)(30,45,37,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=fgi,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, 6599975);