Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13453 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C127 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 155 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 3 generators on 155 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,19)(16,20)(17,22)(18,26)(21,24)(23,27)(25,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,17,18)(14,19,22,26)(16,23,25,24)(20,27,28,21);(2,4)(5,8,9,7)(6,10,11,12)(14,20)(15,18)(16,24)(19,21)(22,28)(23,25)(26,27)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6441);