Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60600 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,20,18)(16,19,23,21)(17,24,27,29)(22,28,25,26);(7,10)(11,13)(14,16)(15,21)(17,24)(18,19)(20,23)(22,28)(25,26)(27,29);(14,17,20,27)(15,22,18,25)(16,24,23,29)(19,26,21,28);(6,8,9,12)(7,11,10,13)(14,18,20,15)(16,19,23,21)(17,25,27,22)(24,26,29,28);(14,19,20,21)(15,23,18,16)(17,26,27,28)(22,29,25,24)305 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53588);