#9230868

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

4570 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)(3,4)(5,6)(7,8)(10,11)(13,21)(15,17)(19,22)(25,26,31,29)(27,35,37,32)(28,38,39,33)(30,34,40,36);(2,6)(3,7)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,24,22,23)(25,27)(26,32)(28,36)(29,35)(30,33)(31,37)(34,39)(38,40);(1,3,5,7)(2,4,6,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,23,22,24)(25,26,31,29)(27,32,37,35)(28,33,39,38)(30,34,40,36);(1,4)(2,7)(3,6)(5,8)(9,11,14,10)(12,15,20,17)(13,16,21,18)(19,24,22,23)(25,28)(26,33)(27,36)(29,38)(30,32)(31,39)(34,37)(35,40);(9,12)(10,15)(11,17)(13,22)(14,20)(16,24)(18,23)(19,21)(25,29,31,26)(27,35,37,32)(28,33,39,38)(30,34,40,36);(9,13,14,21)(10,16,11,18)(12,19,20,22)(15,23,17,24)(25,30)(26,34)(27,33)(28,35)(29,36)(31,40)(32,39)(37,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,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,e^d=egi,f^d=fgi,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, 9138064);