#6578654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573652
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	#3494	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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)(12,23)(13,18)(14,19)(15,22)(17,26)(20,25)(21,31)(24,36)(27,33)(28,30)(29,34)(32,39)(35,38)(37,40);(1,3,5,4)(2,6,8,7)(9,11,14,22)(10,15,19,16)(12,17,25,33)(13,18,29,34)(20,26,23,27)(21,31,28,30)(24,35,38,36)(32,40,37,39);(9,12,14,25)(10,17,19,33)(11,20,22,23)(13,24,29,38)(15,27,16,26)(18,32,34,37)(21,35,28,36)(30,40,31,39);(1,3,5,4)(2,7,8,6)(9,10)(11,15)(12,20)(13,28)(14,19)(16,22)(17,26)(18,30)(21,29)(23,25)(24,37)(27,33)(31,34)(32,38)(35,39)(36,40);(1,4,5,3)(2,6,8,7)(9,13,14,29)(10,18,19,34)(11,21,22,28)(12,24,25,38)(15,30,16,31)(17,32,33,37)(20,35,23,36)(26,39,27,40)

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,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485850);