Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C127 · Schur multiplier (H2) = C24×C4 |
| direct product | C127×#20459 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C22×D4 |
| commutator | C4 | C22×C42×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C22×C127 | C22×C4⋊C4 |
| prime radical | C24×C127 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C2×#117×C127 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2,4,3)(5,6,9,14,11,16,10,15)(7,17,19,26,20,30,24,12)(8,13,22,28,21,27,23,29)(18,31,33,35,32,36,34,25)(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);(5,7,10,19,11,20,9,24)(6,12,15,17,16,26,14,30)(8,18,23,33,21,32,22,34)(13,25,29,31,27,35,28,36);(1,3,4,2)(5,8,11,21)(6,13,16,27)(7,18,20,32)(9,22,10,23)(12,25,26,35)(14,28,15,29)(17,31,30,36)(19,33,24,34);(8,21)(13,27)(18,32)(22,23)(25,35)(28,29)(31,36)(33,34)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13447);