Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22×C4 |
| direct product | C127×#13215 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1961 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C127×#3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2)(3,14)(4,19,23,28)(5,17)(6,11)(7,13)(8,12)(9,22,27,25)(10,15)(16,21,26,31)(18,24)(20,29,32,30)(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);(1,3,15,13,6,18,24,11,7,10,14,2,8,17,5,12)(4,20,29,28,22,26,21,27,23,32,30,19,25,16,31,9);(1,4,7,23)(2,9)(3,16)(5,21,24,31)(6,22,8,25)(10,26)(11,19)(12,28)(13,27)(14,29,15,30)(17,32)(18,20)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6203);