Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#11336 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #127 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C23×C4 | C23×C127 |
| socle | C22×C127 | #399 |
| prime radical | C24×C127 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(5,6,10,14,11,16,9,15)(7,17,21,27,20,28,19,12)(8,13,23,31,25,32,24,30)(18,29,34,26,35,33,22,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,2)(3,4)(5,7,9,19,11,20,10,21)(6,12,14,17,16,27,15,28)(8,22,23,18,25,34,24,35)(13,29,30,36,32,33,31,26);(2,4)(5,8)(6,13)(7,18,20,35)(9,24)(10,23)(11,25)(12,26,27,36)(14,31)(15,30)(16,32)(17,33,28,29)(19,22,21,34)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4324);