Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C27 |
| direct product | C127×#21000 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | C2×#121 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C2×#386 |
| prime radical | C2×#575×C127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G | |
1704 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)(3,7,12,21)(4,14,15,26)(5,10)(6,9)(8,16,23,18)(11,27)(13,22,17,25)(19,24)(20,30)(28,31)(29,32)(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)(2,7,9,21)(4,11)(5,17)(6,12)(8,20,23,32)(10,25,24,22)(13,19)(14,31,26,27)(15,28)(16,30)(18,29);(1,4)(2,8)(3,11)(5,18)(6,15)(7,20)(9,23)(10,26)(12,28)(13,30)(14,24)(16,19)(17,29)(21,32)(22,31)(25,27);(3,12)(7,21)(11,28)(13,17)(20,32)(22,25)(27,31)(29,30)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13988);