#6454867

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
prime radical	#62850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2924 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)(12,25)(13,18)(14,30)(15,20)(16,23)(17,28)(19,33)(22,39)(24,31)(26,38)(27,40)(29,36)(32,37)(34,35);(9,11,14,23,15,24,33,18)(10,16,19,21,20,13,30,31)(12,26,29,40,28,22,32,35)(17,34,36,39,25,27,37,38);(2,5)(6,8)(9,12,15,28)(10,17,20,25)(11,22,24,26)(13,27,16,34)(14,32,33,29)(18,35,23,40)(19,37,30,36)(21,38,31,39);(9,13,15,16)(10,18,20,23)(11,19,24,30)(12,27,28,34)(14,21,33,31)(17,35,25,40)(22,37,26,36)(29,39,32,38);(1,3,4,7)(2,6,5,8)(12,28)(17,25)(22,26)(27,34)(29,32)(35,40)(36,37)(38,39)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,e^2=gi,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, 6362063);