#6567502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535687
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3480	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6474698);