Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#21558 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | #121 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C4×#121 |
| prime radical | C127×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,7,12,8,13,6,23)(3,14,5,11,17,28,16,9)(4,18,21,10,22,26,20,27)(15,24,19,29,31,25,30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,3,6,16,8,17,7,5)(2,9,12,14,13,11,23,28)(4,15,20,30,22,31,21,19)(10,24,26,29,27,25,18,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,22)(9,24)(11,25)(12,26)(13,27)(14,29)(16,30)(17,31)(18,23)(28,32);(1,5,7,17,8,16,6,3)(2,11,12,28,13,9,23,14)(4,19,21,31,22,30,20,15)(10,25,26,32,27,24,18,29)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14546);