Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21877 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C5×C4⋊C4 ◁ G | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(14,15)(16,19)(17,24,25,20)(18,21)(22,28,29,26)(23,27);(6,7)(8,11)(9,10)(12,13)(14,16,18,23)(15,19,21,27)(17,22,25,29)(20,26,24,28);(7,10)(11,13)(14,17)(15,20)(16,22)(18,25)(19,26)(21,24)(23,29)(27,28);(6,8,9,12)(7,11,10,13)(16,23)(19,27)(22,29)(26,28)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14865);