Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10196 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3245 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
546 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)(6,7,10,8)(9,12,11,13)(14,15,19,23)(16,24,26,29)(17,22,18,21)(20,27,28,25);(7,12)(8,13)(9,11)(14,16)(15,20)(17,27)(18,25)(19,26)(21,29)(22,24)(23,28);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,25)(18,27)(19,26)(21,24)(22,29)(23,28)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3184);