#6606292

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
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 ◁ C₂×C₄×D₄ ◁ 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) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6513488);