Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#10782 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #120 |
| commutator | C2×C4 | C2×C42×C127 |
| Frattini, cosocle | C23×C4 | C23×C127 |
| socle | C22×C127 | #386 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G | |
1178 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)(5,6,9,15,11,16,10,14)(7,17,20,28,21,27,19,12)(8,22,24,29,25,30,23,13)(18,26,32,35,33,36,34,31)(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);(5,7,10,19,11,21,9,20)(6,12,14,27,16,28,15,17)(8,18,24,32,25,33,23,34)(13,26,22,35,29,36,30,31);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,22)(15,30)(16,29)(17,31)(19,32)(20,34)(21,33)(27,35)(28,36)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 3770);