#8759313

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

4711 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)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,26)(27,32)(28,34)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,7)(4,5)(6,8)(10,14)(15,17)(16,21)(22,23)(25,27,31,28)(26,32,33,34)(29,38,35,36)(30,39,40,37);(2,5)(6,7)(11,18)(12,20)(15,17)(16,21)(26,33)(27,28)(29,35)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,27)(26,34,33,32)(29,38,35,36)(30,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,29,31,35)(26,30,33,40)(27,36,28,38)(32,37,34,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,30)(26,35)(27,37)(28,39)(29,33)(31,40)(32,38)(34,36)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8666509);