Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #459 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C23×C127 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×C8×C127 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,24)(10,27)(11,16)(12,18)(13,22)(15,26)(19,32)(20,34)(21,28)(23,33)(25,29)(30,35)(31,36)(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);(1,3,4,2)(5,7)(6,12)(8,18)(9,25)(10,19)(11,21)(13,28)(14,32)(15,29)(16,31)(17,24)(20,26)(22,35)(23,36)(27,30)(33,34);(5,8,10,22,11,23,26,9)(6,13,15,14,16,24,27,33)(7,18,19,35,21,36,20,25)(12,28,29,32,31,17,30,34)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6168);