#6574590

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

1876 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)(9,10,14,21)(11,23,26,17)(12,27,24,19)(13,20,30,37)(15,18,16,22)(25,39,40,33)(28,36,29,35)(31,34,32,38);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,15)(13,29)(14,26)(16,24)(18,19)(20,36)(21,23)(22,27)(25,31)(28,30)(32,40)(33,34)(35,37)(38,39);(10,18)(11,24)(12,26)(15,16)(19,27)(20,34)(21,22)(25,29)(28,40)(31,32)(35,36)(37,38);(9,12,14,24)(10,19,21,27)(11,15,26,16)(13,25,30,40)(17,18,23,22)(20,33,37,39)(28,32,29,31)(34,35,38,36);(9,13)(10,20)(11,25)(12,28)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(21,37)(22,38)(23,39)(24,29)(26,40)(27,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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, 6481786);