#6630364

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629034
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

2064 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,c^b=ci,e^b=ei,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537560);