#6578662

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573643
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	#61262	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1899 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)(11,16)(12,26)(13,23)(14,20)(15,21)(17,28)(18,24)(19,31)(22,34)(25,33)(27,35)(29,36)(30,37)(32,39)(38,40);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,27)(13,14)(15,25)(17,34)(18,19)(20,23)(21,33)(22,28)(24,31)(26,35)(29,39)(30,40)(32,36)(37,38);(9,12,15,29)(10,17,21,37)(11,22,25,40)(13,27,24,39)(14,30,31,28)(16,32,33,35)(18,38,23,34)(19,36,20,26);(1,3,5,4)(2,7,8,6)(9,13,15,24)(10,18,21,23)(11,14,25,31)(12,22,29,40)(16,19,33,20)(17,35,37,32)(26,38,36,34)(27,28,39,30);(1,4,5,3)(2,6,8,7)(9,10)(11,23)(12,17)(13,16)(14,19)(15,21)(18,25)(20,31)(22,34)(24,33)(26,28)(27,32)(29,37)(30,36)(35,39)(38,40)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485858);