#6806821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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

1996 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)(11,21)(12,24)(13,19)(14,18)(15,31)(16,20)(17,23)(22,37)(25,35)(26,33)(27,39)(28,34)(29,40)(30,36)(32,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,23)(10,17,18,21)(12,16,26,15)(13,27,28,25)(19,35,34,39)(20,24,31,33)(22,30,38,29)(32,36,37,40);(9,12,16,23,14,26,15,11)(10,17,20,33,18,21,31,24)(13,25,30,38,28,27,29,22)(19,32,36,39,34,37,40,35);(2,6)(5,7)(10,18)(17,21)(19,34)(20,31)(24,33)(32,37)(35,39)(36,40);(3,8)(5,7)(9,13,16,30,14,28,15,29)(10,19,20,36,18,34,31,40)(11,22,12,25,23,38,26,27)(17,32,33,39,21,37,24,35)

Polycyclic presentation:

2^9|b^2=i,c^2=h,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,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, 6714017);