Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C127 · Schur multiplier (H2) = C22×C4 |
| direct product | C127×#12614 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C127 | D4 |
| commutator | C4 | C2×C4×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C8×D4 |
| prime radical | C2×#41×C127 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C127 ◁ C22×C16×C127 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2)(3,4)(5,6,9,15,10,14,22,29,11,16,23,30,8,12,21,25)(7,13,18,27,19,26,32,35,20,28,33,36,17,24,31,34)(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);(2,4)(6,12,16,14)(13,24,28,26)(15,25,30,29)(27,34,36,35);(5,7,9,18,10,19,22,32,11,20,23,33,8,17,21,31)(6,13,15,27,14,26,29,35,16,28,30,36,12,24,25,34)14224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5602);