#9164017

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

4515 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)(3,7)(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26,31,35)(27,36,40,32)(28,33,38,30)(29,34,37,39);(1,2,6,5)(3,8,7,4)(10,15)(12,17)(16,21)(20,22)(25,27)(26,32)(28,37)(29,38)(30,39)(31,40)(33,34)(35,36);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(16,21)(19,24)(25,28,31,38)(26,33,35,30)(27,37,40,29)(32,34,36,39);(1,2,6,5)(3,4,7,8)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,28)(26,30)(27,37)(29,40)(31,38)(32,39)(33,35)(34,36);(1,4,6,8)(2,7,5,3)(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,29,31,37)(26,34,35,39)(27,38,40,28)(30,32,33,36);(1,5,6,2)(3,8,7,4)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,30)(26,28)(27,39)(29,36)(31,33)(32,37)(34,40)(35,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=ghi,f^2=h,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, 9071213);