Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13071 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #466 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1893 |
| prime radical | C127×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
456 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)(3,14,19,39)(4,20,23,38)(5,24)(6,12)(7,31)(8,13)(9,17,34,25)(10,22,37,30)(11,15)(16,47)(18,43,29,44)(21,36,26,40)(27,41)(28,57)(32,58)(33,53,35,45)(42,50)(46,62,56,55)(48,60)(49,59)(51,63)(52,54)(61,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3,15,45,6,29,5,25,8,19,27,53,7,18,28,17)(2,9,24,43,31,33,11,39,13,34,57,44,12,35,41,14)(4,16,46,58,30,49,26,48,23,51,56,64,22,50,21,52)(10,32,55,63,20,54,40,59,37,61,62,47,38,60,36,42);(1,4)(2,10)(3,16)(5,26)(6,30)(7,22)(8,23)(9,32)(11,40)(12,38)(13,37)(14,42)(15,46)(17,52)(18,50)(19,51)(20,31)(21,28)(24,55)(25,48)(27,56)(29,49)(33,54)(34,61)(35,60)(36,41)(39,59)(43,63)(44,47)(45,58)(53,64)(57,62)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6059);