#6445202

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1943 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)(3,6)(4,5)(7,8)(9,10,16,21)(11,17,23,18)(12,24,26,33)(13,19,29,38)(14,30,15,20)(22,32,39,34)(25,40,27,35)(28,37,31,36);(2,5)(6,8)(9,11)(10,17)(12,14)(13,27)(15,26)(16,23)(18,21)(19,35)(20,33)(22,28)(24,30)(25,29)(31,39)(32,37)(34,36)(38,40);(9,12,15,11,16,26,14,23)(10,18,30,33,21,17,20,24)(13,25,31,22,29,27,28,39)(19,34,37,35,38,32,36,40);(9,13)(10,19)(11,22)(12,25)(14,28)(15,31)(16,29)(17,32)(18,34)(20,36)(21,38)(23,39)(24,40)(26,27)(30,37)(33,35);(1,3,4,7)(2,6,5,8)(11,23)(12,26)(17,18)(22,39)(24,33)(25,27)(32,34)(35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=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, 6352398);