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×#57893 |
| 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 | |
5070 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)(5,9)(6,7)(8,10)(11,12,16,17)(13,15,14,18)(19,20)(21,26)(22,23)(24,27)(25,28)(29,33)(30,31)(32,34);(4,7)(8,9)(12,17)(13,14)(19,21,25,23)(20,26,28,22)(24,31,32,29)(27,30,34,33);(11,13,16,14)(12,18,17,15)(19,22)(20,21)(23,28)(24,30)(25,26)(27,29)(31,34)(32,33);(3,4)(5,8)(6,7)(9,10)(11,14,16,13)(12,15,17,18)(19,23,25,21)(20,26,28,22)(24,31,32,29)(27,33,34,30);(3,5,6,10)(4,8,7,9)(11,15)(12,13)(14,17)(16,18)(19,24,25,32)(20,27,28,34)(21,29,23,31)(22,30,26,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008648);