Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11266 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #388 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1442 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,15,12,17,10,16)(8,18,20,13,22,29,21,30)(9,23,25,33,26,32,27,14)(19,35,36,31,37,28,24,34);(6,8,10,21,12,22,11,20)(7,13,16,18,17,30,15,29)(9,24,25,19,26,36,27,37)(14,31,23,28,33,34,32,35);(6,9,10,25,12,26,11,27)(7,14,15,32,17,33,16,23)(8,19,20,24,22,37,21,36)(13,28,18,34,30,35,29,31)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4254);