Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#18991 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C127 |
| socle | C22×C127 | C2×#120 |
| prime radical | C127×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C2×C42×C127 ◁ G | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2)(3,11)(4,13)(5,8)(6,9)(7,10)(12,15)(14,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11979);