#6683265

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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	#60457	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1895 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)(9,10,16,19)(11,17,21,20)(12,14,26,15)(13,18,29,36)(22,32,38,37)(23,33,30,24)(25,28,35,31)(27,34,39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,24)(13,27)(14,23)(15,30)(16,21)(18,34)(19,20)(22,28)(25,37)(26,33)(29,39)(31,38)(32,35)(36,40);(9,12,14,19,16,26,15,10)(11,20,23,33,21,17,30,24)(13,25,28,36,29,35,31,18)(22,37,39,40,38,32,27,34);(2,6)(5,7)(11,21)(17,20)(22,38)(23,30)(24,33)(27,39)(32,37)(34,40);(3,8)(5,7)(9,13)(10,18)(11,22)(12,25)(14,28)(15,31)(16,29)(17,32)(19,36)(20,37)(21,38)(23,39)(24,34)(26,35)(27,30)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,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,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, 6590461);