Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13073 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #466 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#472×C127 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2)(3,4)(5,6,10,15,12,8,11,23)(7,16,20,33,21,32,28,34)(9,24,26,22,17,35,27,14)(13,29,30,19,18,36,31,25)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(5,7,17,19,11,20,26,36,12,21,9,25,10,28,27,29)(6,13,14,32,15,31,24,33,8,18,22,16,23,30,35,34);(2,4)(5,8)(6,12)(7,18)(9,14)(10,15)(11,23)(13,21)(16,19)(17,22)(20,30)(24,27)(25,32)(26,35)(28,31)(29,33)(34,36)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6061);