#6668620

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2664 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)(11,16)(12,18)(13,17)(14,19)(15,20)(21,30)(22,29)(23,31)(24,32)(25,34)(26,35)(27,33)(28,36)(37,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11,15,24)(10,16,20,32)(12,25,26,38)(13,23,14,21)(17,31,19,30)(18,34,35,40)(22,28,37,27)(29,36,39,33);(10,17)(11,21)(13,14)(18,33)(19,20)(22,25)(23,24)(27,28)(30,31)(34,40)(35,36)(37,38);(2,6)(5,7)(12,26)(18,35)(22,37)(25,38)(27,28)(29,39)(33,36)(34,40);(3,8)(5,7)(9,12)(10,18)(11,22)(13,27)(14,28)(15,26)(16,29)(17,33)(19,36)(20,35)(21,25)(23,38)(24,37)(30,34)(31,40)(32,39)

Polycyclic presentation:

2^9|b^2=fi,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, 6575816);