Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12712 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1781 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×2−1+4 ◁ G | |
952 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)(8,16,19,28)(9,20,22,12)(10,14)(11,15)(13,18,27,23)(17,31)(21,35,37,26)(24,29)(25,32)(30,33)(34,36);(7,12)(13,25)(14,26)(15,20)(16,30)(27,34)(28,31)(29,35);(6,8,11,19)(7,13,15,27)(9,17,22,33)(10,23,24,18)(12,25,20,34)(14,28,29,16)(21,36,37,32)(26,31,35,30)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5700);