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×#21075 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #121 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C22×C127 | C2×#121 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,14)(10,25)(11,15)(12,19)(13,24)(17,31)(18,27)(20,30)(22,36)(23,28)(26,32)(29,34)(33,35)(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);(1,3,4,2)(5,7,10,19,11,20,9,18)(6,12,14,16,15,27,25,30)(8,17,23,33,24,34,22,32)(13,26,28,29,21,35,36,31);(5,8,10,23,11,24,9,22)(6,13,14,28,15,21,25,36)(7,17,19,33,20,34,18,32)(12,26,16,29,27,35,30,31);(5,9,11,10)(6,14,15,25)(7,18,20,19)(8,22,24,23)(12,16,27,30)(13,28,21,36)(17,32,34,33)(26,29,35,31)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14063);