#6574960

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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	#61057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1959 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)(13,19)(14,20)(15,21)(16,23)(17,26)(18,30)(22,37)(24,28)(25,29)(27,32)(31,40)(33,35)(34,39)(36,38);(2,5)(6,8)(9,11,15,26)(10,17,21,12)(13,24,33,20)(14,35,28,19)(16,27,37,18)(22,32,23,30)(25,31,39,36)(29,38,34,40);(1,3,4,7)(2,6,5,8)(9,12,15,17)(10,18,21,27)(11,22,26,23)(13,31,33,36)(14,29,28,34)(16,32,37,30)(19,24,35,20)(25,38,39,40);(9,13,15,33)(10,19,21,35)(11,24,26,20)(12,28,17,14)(16,34,37,29)(18,38,27,40)(22,25,23,39)(30,36,32,31);(9,14,15,28)(10,20,21,24)(11,25,26,39)(12,29,17,34)(13,32,33,30)(16,31,37,36)(18,19,27,35)(22,38,23,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482156);