#6519824

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

1911 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:

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

Polycyclic presentation:

2^9|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^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, 6427020);