Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,21)(14,15)(16,18)(17,23)(19,22)(20,24)(25,26,31,35)(27,36,38,32)(28,33,37,30)(29,34,39,40);(2,5)(6,7)(10,15)(12,16)(17,21)(20,22)(25,27,31,38)(26,32,35,36)(28,29,37,39)(30,34,33,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(17,21)(19,24)(25,28,31,37)(26,33,35,30)(27,29,38,39)(32,40,36,34);(1,3,4,8)(2,6,5,7)(9,11,14,18)(10,16,15,12)(13,19,23,24)(17,22,21,20)(25,28)(26,30)(27,29)(31,37)(32,34)(33,35)(36,40)(38,39);(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,29)(26,34)(27,37)(28,38)(30,36)(31,39)(32,33)(35,40);(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,30)(26,28)(27,34)(29,32)(31,33)(35,37)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=hi,e^2=h,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=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068000);