#8254847

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

4388 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8162043);