Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7604 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C43×C5 ◁ G | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,9,11,10,12,8,13)(14,15,17,20,18,21,16,19)(22,23)(24,30,32,27)(25,28)(26,29)(31,36,37,34)(33,35);(7,11)(8,9)(12,13)(22,24,26,32)(23,27,29,30)(25,31,33,37)(28,34,35,36);(7,12)(11,13)(14,16,18,17)(15,19,21,20)(22,25,26,33)(23,28,29,35)(24,31,32,37)(27,34,30,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 592);