Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4902 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)(11,12,16,13)(14,18,15,17)(19,20)(21,25)(22,29)(23,26)(24,27)(28,32)(30,34)(31,33);(4,5)(8,9)(12,13)(14,15)(19,21,24,22)(20,25,27,29)(23,28,31,30)(26,32,33,34);(11,13,16,12)(14,18,15,17)(19,22,24,21)(20,25,27,29)(23,30,31,28)(26,32,33,34);(3,5,7,4)(6,9,10,8)(11,14,16,15)(12,17,13,18)(19,23,24,31)(20,26,27,33)(21,28,22,30)(25,32,29,34);(3,6,7,10)(4,8,5,9)(11,15,16,14)(12,18,13,17)(19,20)(21,29)(22,25)(23,26)(24,27)(28,34)(30,32)(31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013066);