Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#21181 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #121 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C22×C127 | C2×#121 |
| prime radical | C24×C127 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2,4,3)(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,24)(18,32)(19,27)(21,30)(23,29)(25,28)(26,35)(31,34)(33,36)(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)(9,19)(10,20)(11,21)(13,26)(14,17)(15,30)(16,27)(22,31)(23,33)(24,34)(25,35)(28,36)(29,32);(5,8,10,24,11,25,9,23)(6,13,14,22,16,29,15,28)(7,18,20,34,21,35,19,33)(12,26,17,31,27,32,30,36);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,17,27,30)(13,22,29,28)(18,33,35,34)(26,31,32,36)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14169);