Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1415 |
| commutator | C32 | C2×C4×C5 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| prime radical | C22×C5 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ C2×C5×C16 ◁ C2×C5×C32 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 133 points:
(1,2,3,4,5)(6,7,10,16,13,20,37,19)(8,21,23,67,26,71,34,70)(9,27,30,83,33,87,95,86)(11,38,35,103,41,53,105,106)(12,42,36,18,44,59,81,65)(14,45,46,22,50,73,49,76)(15,51,54,97,57,116,56,31)(17,60,58,29,63,92,62,40)(24,78,77,123,79,119,101,129)(25,80,43,69,82,118,102,127)(28,61,88,126,74,66,93,125)(32,98,94,85,100,131,130,121)(39,68,108,120,72,52,104,117)(47,99,110,107,113,133,112,90)(48,75,111,109,114,128,64,89)(55,91,122,132,124,96,84,115);(6,8,13,26)(7,14,20,50)(9,28,33,74)(10,34,37,23)(11,39,41,72)(12,43,44,102)(15,52,57,68)(16,49,19,46)(17,61,63,66)(18,64,65,111)(21,27,71,87)(22,31,76,97)(24,29,79,40)(25,81,82,36)(30,93,95,88)(32,99,100,133)(35,104,105,108)(38,47,53,113)(42,48,59,114)(45,51,73,116)(54,120,56,117)(55,123,124,129)(58,125,62,126)(60,101,92,77)(67,86,70,83)(69,98,127,131)(75,115,128,132)(78,84,119,122)(80,85,118,121)(89,91,109,96)(90,130,107,94)(103,112,106,110)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 530);