#6683249

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

1928 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)(9,10,15,20)(11,16,23,32)(12,24,22,36)(13,17,30,35)(14,19,25,18)(21,31,38,40)(26,39,27,33)(28,37,29,34);(1,3,4,8)(2,5,6,7)(9,11,15,23)(10,16,20,32)(12,14,22,25)(13,27,30,26)(17,33,35,39)(18,24,19,36)(21,28,38,29)(31,37,40,34);(9,12,25,23,15,22,14,11)(10,16,19,36,20,32,18,24)(13,26,29,38,30,27,28,21)(17,31,37,33,35,40,34,39);(2,6)(5,7);(3,8)(5,7)(9,13)(10,17)(11,21)(12,26)(14,28)(15,30)(16,31)(18,34)(19,37)(20,35)(22,27)(23,38)(24,39)(25,29)(32,40)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,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, 6590445);