Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#61489 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#529 ◁ G | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(7,10)(11,13)(14,15)(16,19)(17,20)(18,22)(21,25)(23,27)(24,28)(26,29);(6,7)(8,11)(9,10)(12,13)(14,16)(15,19)(17,24)(18,23)(20,28)(21,26)(22,27)(25,29);(15,20)(16,23)(17,25)(19,26)(21,22)(27,28);(7,10)(11,13);(6,8,9,12)(7,11,10,13)(16,23)(19,27)(24,29)(26,28)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 54477);