Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13556 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C127 | C22⋊C4 |
| commutator | C22 | C22×C16×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C2×C127 | C2×C22⋊C16 |
| prime radical | C2×#41×C127 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C127 ◁ C22×C16×C127 ◁ G | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,4,10,5,11,19,32,6,12,20,33,22,34,44,56,7,9,21,28,23,29,45,51,24,30,46,52,47,53,61,63)(3,13,15,31,16,35,39,54,17,36,40,55,41,57,59,64,14,8,18,25,37,26,42,48,38,27,43,49,58,50,60,62)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3)(2,8)(4,18)(5,16)(6,17)(7,14)(9,13)(10,31)(11,26)(12,27)(15,21)(19,42)(20,43)(22,41)(23,37)(24,38)(25,28)(29,35)(30,36)(32,54)(33,55)(34,50)(39,45)(40,46)(44,60)(47,58)(48,51)(49,52)(53,57)(56,64)(59,61)(62,63);(3,14)(8,13)(15,18)(16,37)(17,38)(25,31)(26,35)(27,36)(39,42)(40,43)(41,58)(48,54)(49,55)(50,57)(59,60)(62,64)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6544);