Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C4 |
| direct product | C127×#13113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C2×D8 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C4×D8 |
| prime radical | C23×C127 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C22×C8×C127 ◁ C2×C4×C8×C127 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20,13,18,29,21)(11,19,24,33,23,16,14,30)(12,25,27,37,26,40,28,32)(17,34,36,39,35,31,38,22)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(2,5)(6,8)(9,11,15,24,13,23,29,14)(10,16,20,30,18,19,21,33)(12,22,27,34,26,39,28,31)(17,32,36,25,35,37,38,40);(1,3,4,7)(2,6,5,8)(9,12,13,26)(10,17,18,35)(11,22,23,39)(14,31,24,34)(15,27,29,28)(16,32,19,37)(20,36,21,38)(25,33,40,30)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6101);