Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13625 |
| 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 | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,4,10,8,13,7,31)(3,14,17,47,19,49,15,45)(5,22,20,57,26,44,25,62)(6,27,21,43,30,40,29,12)(9,32,34,16,36,52,33,54)(11,38,37,50,41,61,59,24)(18,55,51,39,56,64,28,48)(23,42,60,63,53,46,58,35)(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)(5,23)(6,28)(7,17)(8,19)(10,33)(11,39)(12,42)(13,36)(14,44)(16,50)(18,29)(20,58)(21,56)(22,49)(24,54)(25,60)(26,53)(27,35)(30,51)(31,34)(32,61)(37,55)(38,52)(40,63)(41,48)(43,46)(45,62)(47,57)(59,64);(1,4,8,7)(2,10,13,31)(3,15,19,17)(5,20,26,25)(6,21,30,29)(9,33,36,34)(11,37,41,59)(12,27,43,40)(14,45,49,47)(16,32,54,52)(18,28,56,51)(22,57,44,62)(23,58,53,60)(24,38,50,61)(35,46,63,42)(39,55,48,64)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6613);