#9138509

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

4407 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,e^2=g,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, 9045705);