Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C127 · Schur multiplier (H2) = C24×C4 |
| direct product | C127×#20446 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C22×D4 |
| commutator | C4 | C22×C42×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C22×C127 | C2×C4×D4 |
| prime radical | C24×C127 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C22×M16×C127 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 163 points:
(5,6)(7,15,18,28)(8,12)(9,14)(10,13)(11,17,24,21)(16,29,30,35)(19,26)(20,25)(22,27)(23,31,34,33)(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)(6,11)(8,16)(9,21)(10,18)(12,23)(13,24)(14,28)(15,27)(17,22)(19,30)(20,33)(25,35)(26,34)(29,36)(31,32);(5,8,9,20,10,19,22,32)(6,12,14,25,13,26,27,36)(7,16,21,33,18,30,17,31)(11,23,28,35,24,34,15,29);(6,13)(8,19)(11,24)(14,27)(15,28)(16,30)(20,32)(31,33)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13434);