#6577604

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	#3488	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1910 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)(3,7)(4,6)(5,8)(9,10)(11,21)(12,26)(13,16)(14,18)(15,20)(17,29)(19,32)(22,36)(23,25)(24,31)(27,34)(28,38)(30,35)(33,37)(39,40);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,27)(13,14)(15,24)(17,35)(18,21)(19,31)(20,23)(22,29)(25,32)(26,39)(28,40)(30,33)(34,38)(36,37);(1,3,5,4)(2,7,8,6)(9,12,14,28)(10,17,18,37)(11,22,13,30)(15,33,32,29)(16,34,21,39)(19,38,20,26)(23,35,31,36)(24,27,25,40);(1,4,5,3)(2,6,8,7)(9,13)(10,16)(11,14)(12,27)(15,25)(17,36)(18,21)(19,31)(20,23)(22,29)(24,32)(26,34)(28,40)(30,33)(35,37)(38,39);(9,10)(11,23)(12,17)(13,31)(14,18)(15,19)(16,24)(20,32)(21,25)(22,35)(26,29)(27,34)(28,37)(30,36)(33,38)(39,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=ehi,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, 6484800);