Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C27 |
| direct product | C127×#21014 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #386 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C24 | C24×C127 |
| socle | C2×C127 | C2×#386 |
| prime radical | C127×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G | |
1748 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)(3,7,13,20)(4,14,16,8)(5,9)(6,10)(11,18)(12,19,25,30)(15,27,28,21)(17,22)(23,29)(24,26)(31,32)(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)(2,7,10,20)(4,11)(5,12)(6,13)(8,18,14,26)(9,19,22,30)(15,23)(16,24)(17,25)(21,29,27,32)(28,31);(1,4)(2,8)(3,11)(5,15)(6,16)(7,18)(9,21)(10,14)(12,23)(13,24)(17,28)(19,29)(20,26)(22,27)(25,31)(30,32);(1,5,6,17)(2,9,10,22)(3,12,13,25)(4,15,16,28)(7,19,20,30)(8,21,14,27)(11,23,24,31)(18,29,26,32)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14002);