#6502939

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502937
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	#60753	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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:

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

Polycyclic presentation:

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

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, 6410135);