#6683281

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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	#60460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590477);