#6514283

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500482
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

1741 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:

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

Polycyclic presentation:

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

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, 6421479);