Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#20713 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #381 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C2×#381 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G | |
1244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,12,7,11,5,10)(3,13,15,26,16,8,21,23)(4,17,20,24,19,9,18,25)(14,22,28,31,29,27,30,32)(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)(4,14)(5,15)(6,21)(7,16)(9,22)(10,23)(11,13)(12,26)(17,27)(18,28)(19,29)(20,30)(24,31)(25,32);(1,4)(2,9,11,17)(3,14)(5,18)(6,20)(7,19)(8,22,13,27)(10,24,12,25)(15,28)(16,29)(21,30)(23,31,26,32);(1,5,7,6)(2,10,11,12)(3,15,16,21)(4,18,19,20)(8,23,13,26)(9,24,17,25)(14,28,29,30)(22,31,27,32)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13701);