#6670346

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
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

1883 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)(9,10,14,19)(11,20,23,30)(12,25,15,17)(13,18,28,33)(16,21,32,24)(22,35,38,40)(26,34,29,27)(31,36,37,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,27)(14,19)(15,25)(16,21)(18,29)(22,37)(23,30)(24,32)(26,33)(28,34)(31,38)(35,36)(39,40);(9,11,14,23)(10,16,19,32)(12,21,15,24)(13,22,28,38)(17,30,25,20)(18,31,33,37)(26,36,29,39)(27,40,34,35);(1,3,4,7)(2,6,5,8)(9,12,14,15)(10,17,19,25)(11,21,23,24)(13,26,28,29)(16,30,32,20)(18,27,33,34)(22,36,38,39)(31,40,37,35);(2,5)(6,8)(9,13)(10,18)(11,22)(12,26)(14,28)(15,29)(16,31)(17,27)(19,33)(20,35)(21,36)(23,38)(24,39)(25,34)(30,40)(32,37)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^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,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6577542);