#9224332

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

4305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,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, 9131528);