#6670360

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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

1904 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:

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

Polycyclic presentation:

2^9|d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,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, 6577556);