#6574589

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481785);