Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#21484 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | C2×#120 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C4×#120 |
| prime radical | C127×#3025 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C8×C127 ◁ G | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,5,7,10,19,18)(3,8,14,12,13,21,28,27)(4,9,17,15,16,22,30,29)(11,23,26,32,25,20,31,24)(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)(4,11)(5,12)(6,14)(7,13)(9,20)(10,21)(15,24)(16,25)(17,26)(18,27)(19,28)(22,23)(29,32)(30,31);(1,4)(2,9)(3,11)(5,15)(6,17)(7,16)(8,20)(10,22)(12,24)(13,25)(14,26)(18,29)(19,30)(21,23)(27,32)(28,31);(1,5,6,10,7,18,19,2)(3,12,14,21,13,27,28,8)(4,15,17,22,16,29,30,9)(11,24,26,23,25,32,31,20)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14472);