#6519818

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500482
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	#60802	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 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:

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

Polycyclic presentation:

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

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, 6427014);