Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53104 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(5,9)(7,10)(11,12)(13,17)(14,16)(15,18)(19,20)(21,22)(23,31)(24,25)(26,29)(27,32)(28,34)(30,33);(3,4)(5,7)(6,8)(9,10)(11,12,14,16)(13,15,18,17)(20,25)(22,29)(27,31)(30,34);(3,4,6,8)(5,7,9,10)(11,13)(12,15)(14,18)(16,17)(19,21)(20,22)(23,28)(24,26)(25,29)(27,30)(31,34)(32,33);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,26)(21,25)(23,30)(24,29)(27,33)(28,31)(32,34);(3,5,6,9)(4,7,8,10)(19,23,24,32)(20,27,25,31)(21,28,26,33)(22,30,29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000093);