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×#11313 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #388 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1443 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
532 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)(8,17,20,13)(9,21)(10,25)(11,16)(12,15)(14,23)(18,35,37,31)(19,29,26,30)(22,28,36,34)(24,33)(27,32);(6,8,10,19,12,20,11,26)(7,13,25,30,15,17,16,29)(9,22,23,18,24,36,27,37)(14,31,33,28,32,35,21,34);(6,9)(7,14,15,32)(8,18)(10,23)(11,27)(12,24)(13,28,17,34)(16,21,25,33)(19,36)(20,37)(22,26)(29,31,30,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4301);