Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13622 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #357 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C2×C127 | C2×#357 |
| prime radical | C2×#508×C127 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G | |
294 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)(3,14,17,45)(4,10)(5,21)(6,26)(7,30)(8,13)(9,16,33,51)(11,23)(12,28)(15,43,31,47)(18,54,53,64)(19,55)(20,41)(22,40,52,44)(24,60)(25,42)(27,46,50,37)(29,38)(32,48,35,62)(34,61,63,56)(36,57)(39,49)(58,59)(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)(4,15)(5,22)(6,27)(7,31)(8,17)(10,32)(11,37)(12,40)(13,33)(14,42)(16,49)(18,28)(19,56)(20,53)(21,45)(23,51)(24,61)(25,52)(26,34)(29,50)(30,35)(36,54)(38,63)(39,46)(41,44)(43,60)(47,55)(48,59)(57,62)(58,64);(1,4,8,7)(2,10,13,30)(3,15,17,31)(5,19,25,24)(6,20,29,28)(9,32,33,35)(11,36,39,58)(12,26,41,38)(14,43,45,47)(16,48,51,62)(18,27,53,50)(21,55,42,60)(22,56,52,61)(23,57,49,59)(34,44,63,40)(37,54,46,64)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6610);