Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#58157 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(19,20,24,27)(21,25,22,29)(23,26,31,34)(28,32,30,33);(4,5)(6,10)(11,12)(13,14)(15,16)(17,18)(19,21,24,22)(20,25,27,29)(23,30,31,28)(26,33,34,32);(4,5)(8,9)(12,16)(13,17)(19,22,24,21)(20,25,27,29)(23,30,31,28)(26,32,34,33);(3,5,7,4)(6,9,10,8)(11,13,15,17)(12,14,16,18)(19,20)(21,25)(22,29)(23,26)(24,27)(28,32)(30,33)(31,34);(3,6,7,10)(4,8,5,9)(11,14)(12,17)(13,16)(15,18)(19,23,24,31)(20,26,27,34)(21,28,22,30)(25,32,29,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011878);