#6349575

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349529
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	#62805	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2916 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)(11,21,25,31)(12,26,29,33)(13,19,14,18)(16,24,32,23)(17,28,34,27)(22,36,39,30)(35,38,40,37);(1,2,5,8)(3,6,4,7)(9,11,15,25)(10,16,20,32)(12,22,29,39)(13,24,14,23)(17,30,34,36)(18,31,19,21)(26,40,33,35)(27,38,28,37);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,26)(19,33)(20,34)(21,35)(23,37)(24,38)(25,39)(31,40)(32,36);(1,3,5,4)(2,6,8,7);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,29,28)(16,21,32,31)(17,26,34,33)(22,37,39,38)(30,35,36,40)

Polycyclic presentation:

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

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, 6256771);