#9039224

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

4583 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8946420);