#6491821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1930 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:

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

Polycyclic presentation:

2^9|b^2=f,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,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, 6399017);