#9719168

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

4353 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9626364);