#6683274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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

1912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,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,e^d=ei,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, 6590470);