#6576438

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573635
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	#24461	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1866 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)(14,17)(15,32)(16,21)(19,30)(22,39)(23,36)(24,34)(25,37)(26,33)(28,35)(29,31)(38,40);(2,5)(6,8)(9,11,17,26)(10,18,14,33)(12,28,20,36)(13,23,27,35)(15,25,16,34)(19,38,29,39)(21,24,32,37)(22,30,40,31);(9,12,17,20)(10,19,14,29)(11,22,26,40)(13,32,27,21)(15,31,16,30)(18,35,33,23)(24,38,37,39)(25,36,34,28);(1,3,4,7)(2,6,5,8)(9,13)(10,20)(11,23)(12,14)(15,19)(16,29)(17,27)(18,36)(21,31)(22,24)(25,38)(26,35)(28,33)(30,32)(34,39)(37,40);(9,14)(10,17)(11,24)(12,29)(13,30)(15,32)(16,21)(18,34)(19,20)(22,38)(23,36)(25,33)(26,37)(27,31)(28,35)(39,40)

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,g^a=gi,c^b=ch,e^b=ehi,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483634);