#6554676

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
prime radical	#61405	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1674 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=h,e^2=hi,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, 6461872);