Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C28 |
| direct product | C127×#20593 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C22×2−1+4×C127 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×#41×C127 ◁ G | |
3594 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)(5,6,9,14,10,13,22,27)(7,15,17,28,18,11,21,24)(8,12,20,26,19,25,32,35)(16,29,31,36,30,23,33,34)(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);(5,7,10,18)(6,11,13,15)(8,16,19,30)(9,21,22,17)(12,23,25,29)(14,28,27,24)(20,33,32,31)(26,36,35,34);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,25)(14,26)(15,29)(17,31)(18,30)(21,33)(22,32)(24,34)(27,35)(28,36);(8,19)(12,25)(16,30)(20,32)(23,29)(26,35)(31,33)(34,36)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13581);