Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1898 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C2×#41 | C23×C127 |
| prime radical | C2×2−1+4×C127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×2−1+4×C127 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,7,13,8,14,6,12)(3,15,20,45,21,46,19,44)(4,22,25,10,26,38,24,39)(5,27,16,42,30,11,31,33)(9,32,35,28,36,54,41,18)(17,48,51,43,52,63,50,60)(23,57,47,37,29,56,58,40)(34,53,61,64,62,49,55,59)(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);(1,3,16,32)(2,9,33,45)(4,17,47,59)(5,28,7,19)(6,20,30,18)(8,21,31,54)(10,34,56,48)(11,15,13,35)(12,41,27,46)(14,36,42,44)(22,55,40,43)(23,53,25,50)(24,51,29,49)(26,52,58,64)(37,60,38,61)(39,62,57,63);(1,4,8,26)(2,10,14,39)(3,17,21,52)(5,29,30,23)(6,25,7,24)(9,34,36,62)(11,40,27,37)(12,38,13,22)(15,43,46,60)(16,47,31,58)(18,53,28,49)(19,51,20,50)(32,59,54,64)(33,56,42,57)(35,55,41,61)(44,63,45,48)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6089);