Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3036 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,10)(9,12)(11,13)(14,15)(16,23)(17,25)(18,26)(19,22)(20,24)(21,28)(27,29);(14,16)(15,20)(17,21)(18,27)(19,25)(22,29)(23,26)(24,28);(7,10)(8,13)(11,12)(14,17,18,20,19,23,28,29)(15,16,21,24,22,25,26,27)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3444);