#6579440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574086
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61087	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,17)(13,19)(14,20)(15,24)(16,28)(18,31)(22,33)(23,35)(25,34)(26,38)(27,39)(29,36)(30,37)(32,40);(2,5)(6,8)(9,11,13,24,14,26,31,25)(10,15,18,21,20,34,19,38)(12,27,28,40,30,23,29,22)(16,35,17,32,36,39,37,33);(1,3,4,7)(2,6,5,8)(9,12,13,28,14,30,31,29)(10,16,18,17,20,36,19,37)(11,22,24,27,26,40,25,23)(15,32,21,39,34,33,38,35);(9,12,14,30)(10,17,20,37)(11,23,26,27)(13,28,31,29)(15,33,34,32)(16,18,36,19)(21,35,38,39)(22,25,40,24);(9,10)(11,21)(12,16)(13,18)(14,20)(15,25)(17,28)(19,31)(22,39)(23,32)(24,34)(26,38)(27,33)(29,37)(30,36)(35,40)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,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, 6486636);