#9070618

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

4633 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)(9,10,14,15)(11,19,21,16)(12,17,22,20)(13,23,24,18)(25,26,31,35)(27,32,36,38)(28,30,34,39)(29,33,37,40);(3,7)(5,8)(10,15)(12,22)(16,19)(18,23)(25,27,31,36)(26,32,35,38)(28,29,34,37)(30,33,39,40);(1,2)(3,5)(4,6)(7,8)(10,15)(12,22)(16,19)(18,23)(27,36)(29,37)(32,38)(33,40);(9,11)(10,16)(12,23)(13,17)(14,21)(15,19)(18,22)(20,24)(25,28)(26,30)(27,29)(31,34)(32,33)(35,39)(36,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,20)(16,24)(21,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40);(1,2)(3,5)(4,6)(7,8)(9,13,14,24)(10,18,15,23)(11,20,21,17)(12,19,22,16)(25,30)(26,34)(27,33)(28,35)(29,38)(31,39)(32,37)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8977814);