#9143352

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

4584 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^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, 9050548);