Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23×C42 |
| direct product | C5×#14539 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(10,11,13,12)(14,15,18,16)(17,20,21,19)(23,24)(27,28);(7,9)(15,16)(17,21)(22,23,26,24)(25,28,29,27);(6,7)(8,9)(10,12,13,11)(14,16,18,15)(17,20,21,19)(22,24,26,23)(25,28,29,27);(14,17)(15,19)(16,20)(18,21)(22,25)(23,27)(24,28)(26,29)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 7527);