#6544927

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1940 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=i,f^2=i,g^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, 6452123);