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×#53917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6270 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);(3,4,7,9)(5,6,10,8)(12,16)(14,18)(20,24)(21,27)(26,31)(29,33);(3,5,7,10)(4,8,9,6)(11,12,15,16)(13,18,17,14)(22,30)(26,31)(29,33)(32,34);(4,9)(6,8)(12,16)(13,17)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33);(3,6,7,8)(4,5,9,10)(11,13,15,17)(12,14,16,18)(19,21)(20,25)(22,29)(23,27)(24,28)(26,32)(30,33)(31,34);(3,5)(4,8)(6,9)(7,10)(11,14)(12,17)(13,16)(15,18)(19,22)(20,26)(21,29)(23,30)(24,31)(25,32)(27,33)(28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010051);