#6454859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2944 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362055);