#8583313

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8490509);