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×#52349 |
| 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 | |
5048 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)(8,10)(11,12)(13,16)(14,18)(15,17)(19,20)(21,24)(22,25)(23,28)(26,31)(27,29)(30,33)(32,34);(3,4)(5,8)(6,7)(9,10)(11,13,15,18)(12,16,17,14)(20,21)(22,29)(26,28)(30,34);(4,7)(8,10)(11,13)(12,14)(15,18)(16,17)(19,21,24,20)(22,25,29,27)(23,28,31,26)(30,32,34,33);(3,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22,24,29)(20,25,21,27)(23,30,31,34)(26,32,28,33);(3,5)(4,8)(6,9)(7,10)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999427);