#6349069

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349016
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	C₄²	C₂⁵
prime radical	#62793	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256265);