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×#58868 |
| 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 | |
4940 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);(3,4,6,7)(5,9,10,8)(11,12)(13,16)(14,15)(17,18)(19,20,25,22)(21,26,23,27)(24,28,32,30)(29,33,31,34);(4,7)(5,10)(12,14)(13,17)(19,21,25,23)(20,26,22,27)(24,31,32,29)(28,34,30,33);(4,7)(8,9)(11,13,15,17)(12,16,14,18)(19,22,25,20)(21,26,23,27)(24,30,32,28)(29,33,31,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,18)(15,17)(19,23,25,21)(20,27,22,26)(24,31,32,29)(28,34,30,33);(3,5)(4,8)(6,10)(7,9)(11,14,15,12)(13,18,17,16)(19,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,33)(27,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012264);