Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18984 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
990 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)(11,13)(14,15)(16,24)(17,26)(18,21)(19,22)(20,23)(25,28)(27,29);(8,12)(10,13)(14,16)(15,17)(18,25)(19,26)(20,27)(21,24)(22,29)(23,28);(6,7)(8,13)(9,11)(10,12)(14,17)(15,16)(18,24)(19,29)(20,28)(21,25)(22,26)(23,27);(14,18,20,19)(15,21,23,22)(16,25,27,26)(17,24,28,29)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11972);