#8417243

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

4201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 8324439);