#6445785

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
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 ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6352981);