Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#20710 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #381 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C24 | C24×C127 |
| socle | C2×C127 | C2×#381 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G | |
1276 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)(3,11,14,7)(4,15)(5,9)(6,10)(8,17)(12,19,26,23)(13,24,27,20)(16,28)(18,22)(21,29)(25,30,32,31)(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)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,23)(16,25)(17,26)(18,27)(21,30)(22,24)(28,31)(29,32);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,19,11,23)(9,21,22,28)(13,25)(14,26)(18,29)(20,30,24,31)(27,32);(1,5,6,18)(2,9,10,22)(3,13,14,27)(4,16,17,29)(7,20,11,24)(8,21,15,28)(12,25,26,32)(19,30,23,31)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13698);