Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22)(25,26,31,34,30,29,27,32)(28,36,37,39,38,33,35,40);(2,5)(3,7)(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,24,19,22)(25,27,30,31)(26,32,29,34)(28,37,38,35)(33,40,36,39);(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24)(25,28,27,35,30,38,31,37)(26,33,32,39,29,36,34,40);(1,3,4,7)(2,6,5,8)(10,16)(14,23)(15,19)(22,24)(26,29)(32,34)(33,36)(39,40);(2,5)(6,8)(25,29,31,32,30,26,27,34)(28,33,37,40,38,36,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338231);