Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#9525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #386 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C23×C4 | C23×C127 |
| prime radical | C2×#121×C127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G | |
538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,5,12,6,13,15,25)(3,8,7,14,17,10,22,29)(4,18,21,24,20,30,31,28)(9,26,27,32,11,16,23,19)(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);(2,8,13,10)(4,19)(9,23,11,27)(12,14,25,29)(16,31)(18,28,30,24)(20,32)(21,26);(2,9)(3,15)(4,20)(5,17)(8,23)(10,27)(11,13)(12,24)(14,18)(16,21)(19,32)(25,28)(26,31)(29,30)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 2513);