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×#53626 |
| 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 | |
5342 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)(5,7)(6,8)(9,10)(19,20,24,29)(21,26,32,25)(22,27,28,23)(30,31,33,34);(6,10)(8,9)(11,12)(13,17)(14,15)(16,18)(19,21,24,32)(20,25,29,26)(22,33,28,30)(23,34,27,31);(4,5)(8,9)(12,15)(16,17)(19,21)(20,26)(22,33)(23,31)(24,32)(25,29)(27,34)(28,30);(3,5,7,4)(6,9,10,8)(11,12)(13,16)(14,15)(17,18)(19,22,24,28)(20,27,29,23)(21,30,32,33)(25,34,26,31);(3,6)(4,8)(5,9)(7,10)(11,13)(12,16)(14,18)(15,17)(19,23,24,27)(20,28,29,22)(21,31,32,34)(25,33,26,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010447);