#6676526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1784 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,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,g^c=gi,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, 6583722);