Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12608 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #117 |
| commutator | C2×C4 | C2×C42×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | #432 |
| prime radical | C24×C127 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C2×#97×C127 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2)(3,4)(5,6)(7,17,20,28)(8,14)(9,15)(10,16)(11,13)(12,19,25,18)(21,31,34,36)(22,29)(23,27)(24,30)(26,33,35,32)(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);(2,4)(5,7,9,21,10,19,24,33,11,20,23,34,8,18,22,32)(6,12,15,26,16,17,30,31,13,25,27,35,14,28,29,36);(6,13)(7,18,20,19)(9,23)(12,17,25,28)(14,16)(21,33,34,32)(22,24)(26,36,35,31)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5596);