Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13034 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1879 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,6,11,7,13,32,44,8,14,5,12,28,43,29,47)(3,15,18,36,19,41,30,37,20,51,17,48,33,45,26,9)(4,21,23,49,24,42,27,10,25,38,22,39,34,46,31,40)(16,35,53,59,54,60,56,63,55,61,52,64,58,62,57,50)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3)(2,9,14,37)(4,16)(5,26)(6,30)(7,33)(8,20)(10,35,40,61)(11,41,12,45)(13,48,43,36)(15,47,51,44)(17,29)(18,32)(19,28)(21,50,38,63)(22,57)(23,56)(24,58)(25,55)(27,53)(31,52)(34,54)(39,62,49,60)(42,64,46,59);(1,4,8,25)(2,10,14,40)(3,16,20,55)(5,27,6,31)(7,34,28,24)(9,35,37,61)(11,42,12,46)(13,49,43,39)(15,50,51,63)(17,56,18,57)(19,58,33,54)(21,47,38,44)(22,29,23,32)(26,53,30,52)(36,62,48,60)(41,64,45,59)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6022);