Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C28 |
| direct product | C127×#20623 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #379 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C2×#379 |
| prime radical | C2×#585×C127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G | |
3644 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)(3,12)(4,16,19,9)(5,10)(6,20)(7,11)(8,14)(13,27,29,26)(15,23)(17,31,18,25)(21,24)(22,28,30,32)(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)(2,8)(4,13)(5,14)(6,21)(7,15)(9,22)(10,23)(11,24)(12,20)(16,30)(17,28)(18,32)(19,29)(25,27)(26,31);(1,4)(2,9)(3,13)(5,17)(6,18)(7,19)(8,22)(10,25)(11,16)(12,26)(14,28)(15,29)(20,31)(21,32)(23,27)(24,30);(1,5,7,6)(2,10,11,20)(3,14,15,21)(4,17,19,18)(8,23,24,12)(9,25,16,31)(13,28,29,32)(22,27,30,26)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13611);