Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1804 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7)(8,19,23,38)(9,24)(10,15)(11,31)(12,18)(13,21,37,54)(14,26)(16,29)(17,30)(20,48,51,69)(22,36,32,43)(25,57)(27,58)(28,44)(33,39)(34,42)(35,64,67,55)(40,62)(41,61)(45,60)(46,59,65,50)(47,63,68,53)(49,66,52,56);(6,8,12,23)(7,13)(9,20,27,51)(10,21,30,54)(11,32,34,22)(14,35)(15,19)(16,43)(17,38)(18,37)(24,56)(25,49,28,52)(26,59,61,50)(29,63,60,53)(31,47)(33,64,62,55)(36,45)(39,65)(40,46)(41,67)(42,68)(44,69)(48,57)(58,66);(6,9,12,27)(7,14,18,41)(8,20,23,51)(10,28,30,25)(11,33,34,62)(13,35,37,67)(15,40,17,39)(16,44,45,57)(19,46,38,65)(21,52,54,49)(22,55,32,64)(24,42,58,31)(26,60,61,29)(36,48,43,69)(47,56,68,66)(50,63,59,53)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5815);