Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C24×C4 |
| direct product | C127×#21080 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | C2×#121 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C2×#388 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,10,7,11,5,21)(3,12,16,23,14,26,15,8)(4,17,20,32,19,9,18,24)(13,27,30,22,28,25,29,31)(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)(2,8,11,23)(4,13)(5,15)(6,16)(7,14)(9,22,17,31)(10,12,21,26)(18,29)(19,28)(20,30)(24,25,32,27);(1,4,5,18,7,19,6,20)(2,9,10,24,11,17,21,32)(3,13,15,29,14,28,16,30)(8,22,12,25,23,31,26,27);(3,14)(8,23)(12,26)(13,28)(15,16)(22,31)(25,27)(29,30)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14068);