#6502943

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

1930 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,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, 6410139);