#6445784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437931
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2890 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6352980);