#6681463

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,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, 6588659);