#6580823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,41)(28,39)(29,40)(30,36)(31,44)(32,43)(33,37)(34,38)(35,42)(45,48)(46,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,31)(22,34)(24,39)(25,42)(28,30)(33,35)(36,41)(37,44)(40,43)(45,46);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19)(18,24)(20,32)(21,30)(22,28)(23,36)(25,43)(26,39)(27,34)(29,33)(31,46)(35,45)(37,40)(38,41)(42,48)(44,47);(1,4,5,3)(2,8,6,7)(19,28)(20,31)(22,34)(24,39)(25,44)(26,38)(27,30)(29,32)(33,35)(36,41)(37,42)(40,43);(17,20,21,33)(18,25,23,37)(19,29,30,32)(22,35,34,31)(24,40,36,43)(26,42,38,44)(27,45,28,46)(39,47,41,48)

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488019);