Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60576 |
| 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 | |
2060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(15,20)(16,24)(17,19)(21,28)(25,26)(27,29);(7,10)(11,13)(14,15)(16,21)(17,22)(18,20)(19,23)(24,29)(25,27)(26,28);(14,16,18,25)(15,21,20,27)(17,24,19,26)(22,29,23,28);(6,8,9,12)(7,11,10,13)(14,17,18,19)(15,22,20,23)(16,24,25,26)(21,29,27,28);(15,20)(21,27)(22,23)(28,29)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53564);