Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#12658 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C2×C42 |
| commutator | C2×C4 | C2×C42×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C2×#97 |
| prime radical | C23×C127 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 147 points:
(1,2,3,4)(6,11,13,19)(7,15)(9,18)(12,20,14,16)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(2,4)(5,6,9,14,8,11,15,20,10,13,18,12,17,19,7,16);(5,7,8,9,10,15,17,18)(6,12,19,20,13,14,11,16)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5646);