#9164107

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

4651 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)(12,15)(13,18)(14,20)(16,21)(19,23)(22,24)(25,26)(27,35)(28,30)(29,32)(31,34)(33,40)(36,37)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,19)(13,20)(15,22)(16,23)(17,18)(21,24)(27,29)(32,35)(36,38)(37,39);(1,3,5,4)(2,6,7,8)(9,12,13,21)(10,15,18,16)(11,19,20,24)(14,22,17,23)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,37,40,39);(3,4)(6,8)(9,12)(10,16)(11,19)(13,21)(14,23)(15,18)(17,22)(20,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,39,35,37);(1,4,5,3)(2,6,7,8)(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,22)(23,24)(25,29,31,27)(26,32,34,35)(28,36,33,38)(30,37,40,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,19)(13,20)(14,18)(15,23)(16,22)(21,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,36)(35,38)

Polycyclic presentation:

2^9|c^2=gi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9071303);