Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C27 |
| direct product | C127×#21006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#121 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G | |
1654 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)(3,4)(5,6,10,16,11,15,9,14)(7,12,20,27,19,28,18,26)(8,21,22,13,23,29,24,30)(17,31,32,25,34,35,33,36)(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);(5,7,9,20,11,19,10,18)(6,12,16,26,15,28,14,27)(8,17,22,33,23,34,24,32)(13,25,21,35,30,36,29,31);(2,4)(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,29)(15,30)(16,21)(18,32)(19,34)(20,33)(26,35)(27,31)(28,36);(5,9,11,10)(6,14,15,16)(7,18,19,20)(8,22,23,24)(12,26,28,27)(13,29,30,21)(17,32,34,33)(25,35,36,31)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13994);