#6669263

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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	#61064	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1967 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,18)(14,19)(15,21)(16,20)(17,25)(22,26)(23,27)(24,28)(29,33)(30,34)(31,35)(32,36)(37,39)(38,40);(2,5)(3,7)(9,11)(10,12)(13,17)(14,30)(15,23)(16,24)(18,25)(19,34)(20,28)(21,27)(22,31)(26,35)(29,38)(32,37)(33,40)(36,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,25)(14,26)(15,27)(16,28)(18,24)(19,30)(20,23)(22,35)(29,34)(31,39)(32,40)(33,38)(36,37);(2,5)(6,8)(9,13,16,15)(10,18,20,21)(11,17,24,23)(12,25,28,27)(14,31,32,29)(19,35,36,33)(22,37,38,30)(26,39,40,34);(9,14,16,32)(10,19,20,36)(11,22,24,38)(12,26,28,40)(13,29,15,31)(17,30,23,37)(18,33,21,35)(25,34,27,39)

Polycyclic presentation:

2^9|c^2=f,d^2=i,e^2=i,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,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, 6576459);