#6730802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=gi,e^2=g,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,h^c=hi,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, 6637998);