#6494244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473158
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

1742 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)(9,10,13,19,15,18,14,30)(11,20,23,32,22,33,29,16)(12,24,27,35,26,38,28,17)(21,31,37,36,25,34,39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,25)(13,29)(14,23)(15,22)(17,34)(18,32)(19,33)(20,30)(21,26)(24,36)(27,37)(28,39)(31,35)(38,40);(2,5)(6,8)(9,12,14,28,15,26,13,27)(10,17,19,24,18,35,30,38)(11,21,23,37,22,25,29,39)(16,31,33,40,32,34,20,36);(2,5)(6,8)(11,22)(12,26)(16,32)(17,35)(20,33)(23,29)(24,38)(27,28);(1,3,4,7)(2,6,5,8)(10,18)(16,32)(17,35)(19,30)(20,33)(24,38)(31,34)(36,40)

Polycyclic presentation:

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

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, 6401440);