#6491809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470409
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1935 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6399005);