Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#11935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#102 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| prime radical | C23×C127 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G | |
302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 167 points:
(2,5)(3,7)(6,8)(9,10,13,19,14,20,28,34,15,17,29,35,27,36,12,18)(11,21,23,31,24,37,22,38,25,39,26,16,30,32,40,33)(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);(1,2,4,6)(3,8,7,5)(9,11)(10,16)(12,26)(13,22)(14,30)(15,25)(17,31)(18,37)(19,21)(20,38)(23,28)(24,27)(29,40)(32,34)(33,36)(35,39);(10,17)(16,31)(18,34)(19,35)(20,36)(21,39)(32,37)(33,38)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4923);