Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12609 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1749 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×#97 ◁ G | |
444 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)(8,16,18,12)(9,14)(10,15)(11,13)(17,26,20,24)(19,30,33,25)(21,27)(22,29)(23,28)(31,35)(32,37,34,36);(6,8,9,19,10,20,22,34,11,18,21,33,23,17,31,32)(7,12,14,25,15,26,29,37,13,16,27,30,28,24,35,36);(8,17,18,20)(9,21)(12,24,16,26)(14,27)(19,34,33,32)(22,31)(25,37,30,36)(29,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5597);