#6590289

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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)(11,14)(12,16)(13,15)(18,23)(20,32)(24,38)(26,27)(28,43)(29,46)(33,47)(34,35);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,18,21,23)(19,28,30,43)(20,25,32,39)(22,26,36,27)(24,37,38,31)(29,45,46,41)(33,48,47,40)(34,42,35,44);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(17,19,21,30)(18,24,23,38)(20,33,32,47)(22,37,36,31)(25,41,39,45)(26,28,27,43)(29,34,46,35)(40,44,48,42);(9,12,13,11)(10,16,15,14)(17,20)(18,25)(19,29)(21,32)(22,34)(23,39)(24,40)(26,42)(27,44)(28,45)(30,46)(31,47)(33,37)(35,36)(38,48)(41,43);(1,4,5,3)(2,8,6,7)(18,26)(20,34)(22,36)(23,27)(24,28)(29,47)(31,37)(32,35)(33,46)(38,43)(40,48)(42,44)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6497485);