Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C32 · Schur multiplier (H2) = C2 |
| direct product | C5×#7514 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C32 | D4 |
| commutator | C4 | C2×C5×C32 |
| Frattini, cosocle | C2×C32 | C22×C5 |
| socle | C2×C5 | C22⋊C32 |
| prime radical | C5×#41 | C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C32 ◁ 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)(6,7,10,17,11,18,33,52,12,19,34,53,37,56,80,101,13,20,35,54,38,57,81,102,40,59,83,104,85,106,121,129,14,21,36,55,39,58,82,103,41,60,84,105,86,107,122,130,9,16,29,48,30,49,74,96,31,50,75,97,76,98,117,127)(8,22,24,61,25,62,67,108,26,63,68,109,69,110,114,131,23,46,64,51,65,92,111,99,66,93,112,100,113,125,132,128,27,47,32,77,70,94,78,118,71,95,79,119,115,126,120,133,28,15,42,43,72,44,87,89,73,45,88,90,116,91,123,124);(6,8)(7,15)(9,28)(10,32)(11,25)(12,26)(13,23)(14,27)(16,47)(17,51)(18,44)(19,45)(20,22)(21,46)(24,36)(29,64)(30,72)(31,73)(33,78)(34,79)(35,42)(37,69)(38,65)(39,70)(40,66)(41,71)(43,55)(48,61)(49,94)(50,95)(52,99)(53,100)(54,77)(56,91)(57,62)(58,92)(59,63)(60,93)(67,82)(68,84)(74,111)(75,112)(76,116)(80,120)(81,87)(83,88)(85,113)(86,115)(89,103)(90,105)(96,108)(97,109)(98,126)(101,128)(102,118)(104,119)(106,110)(107,125)(114,122)(117,132)(121,123)(124,130)(127,131)(129,133)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 502);