#6502835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502828
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1967 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6410031);