Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12917 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C127 | C2×D4 |
| commutator | C2×C4 | C22×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C8×D4 |
| prime radical | C2×#41×C127 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C127 ◁ C22×C16×C127 ◁ G | |
384 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)(5,6,11,17,12,16,29,34,9,15,25,18,10,26,28,35)(7,8,21,24,22,23,14,32,19,13,30,31,20,27,33,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(2,4)(5,7,11,21,12,22,29,14,9,19,25,30,10,20,28,33)(6,13,17,31,16,27,34,36,15,8,18,24,26,23,35,32);(5,8)(6,14)(7,18)(9,13)(10,27)(11,24)(12,23)(15,33)(16,30)(17,19)(20,34)(21,26)(22,35)(25,31)(28,36)(29,32)9652 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5905);