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×#56615 |
| 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 | |
5366 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)(8,10)(11,12,14,16)(13,17,18,15)(19,20)(21,23)(22,26)(24,32)(25,28)(27,33)(29,34)(30,31);(5,9)(8,10)(13,18)(15,17)(19,21,25,22)(20,26,28,23)(24,29,33,30)(27,34,32,31);(3,4,6,7)(5,8,9,10)(11,13)(12,15)(14,18)(16,17)(19,22,25,21)(20,26,28,23)(24,30,33,29)(27,34,32,31);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23,25,26)(20,21,28,22)(24,31,33,34)(27,29,32,30);(3,5)(4,8)(6,9)(7,10)(19,24)(20,27)(21,29)(22,30)(23,31)(25,33)(26,34)(28,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006862);