#9143471

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

4397 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=gh,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, 9050667);