#6552479

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459675);