#6513607

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

1907 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)(11,20)(12,26)(13,17)(14,19)(15,24)(16,30)(18,21)(22,32)(23,33)(25,36)(27,34)(28,29)(31,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,21,23,14,25,13,24)(10,15,17,36,19,33,18,20)(12,27,30,39,31,22,29,38)(16,34,26,37,28,32,35,40);(2,5)(3,7)(9,12)(10,16)(11,22)(13,29)(14,31)(15,32)(17,26)(18,35)(19,28)(20,37)(21,30)(23,38)(24,39)(25,27)(33,34)(36,40);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,28)(13,18)(14,19)(16,31)(17,21)(20,24)(22,34)(23,36)(25,33)(26,29)(27,32)(30,35)(37,38)(39,40);(2,5)(6,8)(9,13,14,21)(10,17,19,18)(11,23,25,24)(12,29,31,30)(15,20,33,36)(16,26,28,35)(22,38,27,39)(32,37,34,40)

Polycyclic presentation:

2^9|b^2=fi,d^2=hi,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,d^c=dh,g^c=gi,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, 6420803);