Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5374 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)(12,15)(16,18)(19,20,25,30)(21,27,22,26)(23,28,29,24)(31,32,33,34);(4,7)(8,10)(13,17)(16,18)(19,21,25,22)(20,26,30,27)(23,33,29,31)(24,34,28,32);(3,4)(5,8)(6,7)(9,10)(11,12,14,15)(13,16,17,18)(19,22,25,21)(20,27,30,26)(23,33,29,31)(24,34,28,32);(3,5)(4,8)(6,9)(7,10)(19,23,25,29)(20,28,30,24)(21,31,22,33)(26,34,27,32);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,17)(15,18)(19,24,25,28)(20,29,30,23)(21,32,22,34)(26,33,27,31)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012914);