Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#20673 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | C2×#117 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C2×#380 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×M16×C127 ◁ G | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,10,7,11,5,20)(3,12,16,23,14,27,15,8)(4,17,18,24,19,9,21,25)(13,28,30,32,29,26,31,22)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,3,5,15,7,14,6,16)(2,8,10,12,11,23,20,27)(4,13,18,30,19,29,21,31)(9,22,24,26,17,32,25,28);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,17)(12,26)(14,29)(15,30)(16,31)(20,25)(23,32)(27,28);(3,14)(8,23)(12,27)(13,29)(15,16)(22,32)(26,28)(30,31)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13661);