Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10356 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
452 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)(8,12,13,10)(14,15)(16,18)(17,20)(19,21)(22,23,25,28)(24,32,34,37)(26,31,27,30)(29,33,36,35);(6,8,9,13)(7,10,11,12)(23,28)(24,33)(26,27)(29,32)(34,35)(36,37);(15,18)(16,21)(19,20)(22,24,25,34)(23,29,28,36)(26,33,27,35)(30,37,31,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3344);