Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13039 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #442 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×#97×C127 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(2,4)(5,6,11,17,21,36,9,28,13,19,10,30,12,18,29,16)(7,20,22,34,31,15,27,33,24,35,26,14,23,32,8,25)(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);(1,2,3,4)(5,7,21,23,13,24,12,31)(6,14,18,25,19,34,36,33)(8,9,26,10,27,29,22,11)(15,17,35,16,32,30,20,28);(5,8)(6,15)(7,9)(10,23)(11,31)(12,22)(13,27)(14,17)(16,25)(18,35)(19,32)(20,36)(21,26)(24,29)(28,33)(30,34)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6027);