Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#12720 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #410 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C2×#41 | C23×C127 |
| socle | C2×C127 | #1782 |
| prime radical | C127×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C42×C127 ◁ C4×#41×C127 ◁ G | |
694 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)(3,12)(4,17,19,9)(5,10)(6,11)(7,14,23,13)(8,18,25,32)(15,24)(16,28)(20,26)(21,31)(22,30,27,29)(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)(4,14)(5,15)(6,16)(8,24)(9,12)(10,22)(11,23)(13,19)(17,28)(18,30)(20,31)(21,25)(26,27)(29,32);(1,4)(2,8,11,25)(3,13)(5,18)(6,19)(7,21,23,24)(9,10,17,26)(12,27,28,22)(14,16)(15,29)(20,32)(30,31)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5708);