#6585821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574063
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	#28502	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) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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=eh,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, 6493017);