#6492405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470804
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 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)(12,23)(13,27)(14,17)(15,22)(16,26)(18,28)(20,35)(21,34)(24,33)(25,32)(29,40)(30,38)(31,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,22,14,21,28,38)(10,15,18,19,17,30,27,34)(12,24,25,40,26,20,36,37)(16,31,32,35,23,29,39,33);(2,5)(6,8)(9,12,13,25,14,26,28,36)(10,16,18,32,17,23,27,39)(11,20,22,37,21,24,38,40)(15,29,19,33,30,31,34,35);(2,5)(6,8)(11,21)(15,30)(19,34)(20,24)(22,38)(29,31)(33,35)(37,40);(1,3,4,7)(2,6,5,8)(10,17)(15,30)(16,23)(18,27)(19,34)(29,31)(32,39)(33,35)

Polycyclic presentation:

2^9|b^2=f,c^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 6399601);