Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#11320 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #124 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C23×C4 | C23×C127 |
| socle | C22×C127 | #397 |
| prime radical | C24×C127 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(2,3)(5,6)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,25,22)(16,31,19,30)(20,26)(21,27)(23,29)(24,34,35,33)(32,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,2)(3,4)(5,7,9,22,10,18,23,17)(6,11,13,28,14,25,29,15)(8,19,20,33,21,16,32,34)(12,24,26,31,27,35,36,30);(1,3,4,2)(5,8,9,20,10,21,23,32)(6,12,13,26,14,27,29,36)(7,16,17,33,18,19,22,34)(11,24,15,30,25,35,28,31)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4308);