#6590667

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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)(11,14)(12,16)(13,15)(18,23)(24,38)(25,39)(26,36)(27,43)(40,47)(41,48)(42,44);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,18,21,23)(19,27,29,43)(20,25,33,39)(22,36,35,26)(24,37,38,30)(28,41,46,48)(31,40,45,47)(32,42,34,44);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,19,21,29)(18,24,23,38)(20,31,33,45)(22,37,35,30)(25,41,39,48)(26,43,36,27)(28,34,46,32)(40,42,47,44);(9,12,13,11)(10,16,15,14)(17,20,21,33)(18,25,23,39)(19,28,29,46)(22,32,35,34)(24,40,38,47)(26,44,36,42)(27,41,43,48)(30,31,37,45);(1,4,5,3)(2,6,8,7)(18,26)(19,29)(20,32)(22,35)(23,36)(24,27)(25,39)(28,45)(31,46)(33,34)(38,43)(40,47)

Polycyclic presentation:

2^9|b^2=fg,c^2=f,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497863);