#8874938

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

4443 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)(9,10,14,13)(11,17,18,15)(12,20,22,16)(19,23,21,24)(27,33)(31,34)(35,39)(38,40);(1,2,4,6)(3,5,7,8)(10,13)(15,17)(16,20)(23,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(1,3)(2,5)(4,7)(6,8)(11,18)(15,17)(19,21)(23,24)(26,30)(31,34)(32,36)(38,40);(9,11,14,18)(10,15,13,17)(12,21,22,19)(16,24,20,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,3)(2,5)(4,7)(6,8)(9,13,14,10)(11,17,18,15)(12,20,22,16)(19,24,21,23)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8782134);