#6683270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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

1962 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)(11,20)(12,26)(13,30)(14,19)(15,18)(16,24)(17,28)(21,38)(22,23)(25,33)(27,32)(29,36)(31,35)(34,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11,14,24)(10,16,19,20)(12,27,29,40)(13,25,15,22)(17,34,35,38)(18,33,30,23)(21,31,39,28)(26,37,36,32);(9,12,15,31,14,29,13,28)(10,17,30,36,19,35,18,26)(11,21,22,27,24,39,25,40)(16,32,23,38,20,37,33,34);(2,6)(5,7)(9,13,14,15)(10,18,19,30)(11,22,24,25)(12,28,29,31)(16,23,20,33)(17,26,35,36)(21,27,39,40)(32,38,37,34);(3,8)(5,7)(9,10)(11,23)(12,17)(13,18)(14,19)(15,30)(16,25)(20,22)(21,38)(24,33)(26,28)(27,37)(29,35)(31,36)(32,40)(34,39)

Polycyclic presentation:

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

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, 6590466);