Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C127 · Schur multiplier (H2) = C27 |
| direct product | C127×#20360 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C22×#95 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×#41×C127 ◁ G | |
1996 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)(3,11,13,7)(4,8)(5,10)(6,9)(12,23,25,18)(14,24,27,19)(15,20)(16,21)(17,22)(26,31,32,29)(28,30)(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);(1,3,5,14,6,13,17,27)(2,7,10,19,9,11,22,24)(4,12,16,26,15,25,28,32)(8,18,21,29,20,23,30,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,18)(9,20)(10,21)(11,23)(13,25)(14,26)(17,28)(19,29)(22,30)(24,31)(27,32);(4,15)(8,20)(12,25)(16,28)(18,23)(21,30)(26,32)(29,31)10414 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13348);