Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11032 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #392 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1438 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1252 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)(6,7,11,15,12,16,10,17)(8,18,20,13,21,30,27,29)(9,22,24,32,26,14,25,33)(19,34,23,28,36,31,35,37);(6,8)(7,13)(9,23)(10,27)(11,20)(12,21)(14,31)(15,30)(16,29)(17,18)(19,25)(22,34)(24,36)(26,35)(28,32)(33,37);(6,9)(7,14)(8,19)(10,24)(11,25)(12,26)(13,28)(15,32)(16,22)(17,33)(18,31)(20,35)(21,36)(23,27)(29,37)(30,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4020);