#6502945

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1962 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=i,e^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, 6410141);