#6519821

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	#60559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1900 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,27)(22,36)(23,31)(25,34)(28,33)(29,32)(30,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,30)(18,21)(19,34)(20,31)(26,35)(27,37)(28,39)(29,38)(32,40)(33,36);(2,5)(3,7)(9,12,14,27,15,29,13,28)(10,17,18,26,20,33,19,32)(11,22,24,37,25,38,23,39)(16,30,21,35,31,36,34,40);(1,3,4,7)(2,6,5,8);(2,5)(6,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,31,34)(17,32,33,26)(22,37,38,39)(30,40,36,35)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=i,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, 6427017);