Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C22×C42 |
| direct product | C127×#21985 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C127 | C22×D4 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C24×C127 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C22×C42×C127 ◁ G | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 155 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 155 points:
(1,2,3,4)(6,9)(7,11)(13,14)(15,22)(16,24)(17,21)(18,19)(20,25)(23,28)(26,27)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155);(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,19,21,22)(16,25,26,23)(20,24,28,27);(6,9)(10,12)(13,16,18,25,17,26,15,23)(14,20,22,24,21,28,19,27);(5,7,8,11)(6,10,9,12)(14,21)(19,22)(20,28)(24,27)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14973);