#6673583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3494	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,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, 6580779);