#6814810

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 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:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722006);