#6576104

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	#24460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1885 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6483300);