#6585833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574075
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	#61410	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,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, 6493029);