#6519829

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,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^d=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, 6427025);