#6685496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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

1801 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,25)(13,18)(14,17)(15,20)(16,23)(19,26)(22,38)(24,36)(27,34)(28,40)(29,33)(30,37)(31,35)(32,39);(2,5)(3,7)(9,11,14,23)(10,16,17,21)(12,26,24,15)(13,28,29,27)(18,34,33,40)(19,25,20,36)(22,31,39,30)(32,35,38,37);(9,12,26,11,14,24,15,23)(10,16,20,36,17,21,19,25)(13,27,31,22,29,28,30,39)(18,32,37,40,33,38,35,34);(2,5)(6,8)(10,17)(16,21)(18,33)(19,20)(25,36)(32,38)(34,40)(35,37);(1,3,4,7)(2,6,5,8)(9,13,14,29)(10,18,17,33)(11,22,23,39)(12,27,24,28)(15,30,26,31)(16,32,21,38)(19,35,20,37)(25,34,36,40)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=f,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,h^c=hi,h^e=hi

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, 6592692);