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×#54688 |
| 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 | |
5344 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);(4,7)(5,9)(11,12,16,14)(13,17,15,18)(19,20)(21,26)(22,27)(23,31)(24,25)(28,32)(29,30)(33,34);(3,4,6,7)(5,10,9,8)(12,14)(13,15)(20,25)(22,26)(28,31)(30,33);(3,4,6,7)(5,8,9,10)(11,13,16,15)(12,17,14,18)(19,21,24,27)(20,26,25,22)(23,29,32,34)(28,33,31,30);(3,4)(5,8)(6,7)(9,10)(11,14,16,12)(13,17,15,18)(19,22,24,26)(20,27,25,21)(23,30,32,33)(28,34,31,29);(3,5)(4,8)(6,9)(7,10)(11,15,16,13)(12,17,14,18)(19,23,24,32)(20,28,25,31)(21,29,27,34)(22,30,26,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010768);