#6552868

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

1735 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)(11,21,23,32,25,31,24,16)(12,26,27,34,28,17,29,33)(22,36,38,40,39,35,37,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,24,14,25,15,23)(10,16,19,21,20,32,18,31)(12,22,27,38,28,39,29,37)(17,30,34,35,26,40,33,36);(2,5)(6,8)(9,12,13,27,14,28,15,29)(10,17,19,34,20,26,18,33)(11,22,24,38,25,39,23,37)(16,30,21,35,32,40,31,36);(1,3)(2,6)(4,7)(5,8)(9,13,14,15)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,32,31)(17,33,26,34)(22,37,39,38)(30,35,40,36);(3,7)(6,8)(9,13,14,15)(10,18,20,19)(11,24,25,23)(12,27,28,29)(16,31,32,21)(17,33,26,34)(22,38,39,37)(30,36,40,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,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, 6460064);