#9034326

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

4430 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:

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

Polycyclic presentation:

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

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, 8941522);