#8936589

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

4382 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8843785);