Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7539 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D32 |
| commutator | C32 | C2×C4×C5 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| socle | C2×C5 | #1415 |
| prime radical | C5×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ C2×C5×C16 ◁ C2×C5×C32 ◁ G | |
88 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)(6,7,10,17,14,20,13,45)(8,21,23,64,26,67,46,66)(9,27,30,75,34,78,33,93)(11,37,35,95,41,98,40,101)(12,42,36,63,44,103,39,19)(15,47,48,69,52,106,51,22)(16,53,55,89,57,88,105,31)(18,59,58,85,61,84,97,29)(24,71,70,125,72,126,99,117)(25,73,43,65,74,122,104,123)(28,60,79,119,82,121,94,120)(32,90,86,128,92,132,87,77)(38,54,96,114,100,116,102,115)(49,91,107,130,110,133,109,81)(50,111,108,129,112,68,62,80)(56,83,76,113,118,131,127,124);(6,8)(7,15)(9,28)(10,23)(11,38)(12,43)(13,46)(14,26)(16,54)(17,48)(18,60)(19,62)(20,52)(21,27)(22,31)(24,29)(25,39)(30,79)(32,91)(33,94)(34,82)(35,96)(36,74)(37,49)(40,102)(41,100)(42,50)(44,104)(45,51)(47,53)(55,114)(56,117)(57,116)(58,119)(59,70)(61,121)(63,108)(64,75)(65,90)(66,93)(67,78)(68,124)(69,89)(71,76)(72,85)(73,77)(80,83)(81,87)(84,99)(86,130)(88,106)(92,133)(95,107)(97,120)(98,110)(101,109)(103,112)(105,115)(111,113)(118,125)(122,128)(123,132)(126,127)(129,131)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 527);