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×#58243 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4870 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);(11,12,17,18)(13,14,15,16)(19,20,23,24)(21,27,29,25)(22,30,31,26)(28,32,34,33);(3,4)(5,6)(7,8)(9,10)(11,13,17,15)(12,14,18,16)(20,24)(22,31)(25,27)(28,34);(4,8)(5,9)(11,14,17,16)(12,13,18,15)(19,21,23,29)(20,25,24,27)(22,28,31,34)(26,32,30,33);(3,5,7,9)(4,6,8,10)(11,15,17,13)(12,14,18,16)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34);(3,6)(4,9)(5,8)(7,10)(11,16,17,14)(12,13,18,15)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011797);