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×#57787 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5016 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);(3,4,7,5)(6,9,10,8)(11,12,16,17)(13,15,14,18)(19,20,25,22)(21,23,26,29)(24,27,33,30)(28,31,32,34);(4,5)(8,9)(12,17)(13,14)(19,21,25,26)(20,23,22,29)(24,32,33,28)(27,34,30,31);(11,13,16,14)(12,18,17,15)(19,22,25,20)(21,23,26,29)(24,30,33,27)(28,31,32,34);(3,5,7,4)(6,9,10,8)(11,14,16,13)(12,15,17,18)(19,23)(20,26)(21,22)(24,31)(25,29)(27,32)(28,30)(33,34);(3,6,7,10)(4,8,5,9)(11,15)(12,13)(14,17)(16,18)(19,24)(20,27)(21,28)(22,30)(23,31)(25,33)(26,32)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008653);