Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2054 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,8,14)(3,15,20,50)(4,21,27,57)(5,12,30,47)(6,11,31,46)(7,32,29,13)(9,34,38,17)(10,39,44,24)(16,51,55,35)(18,37,28,48)(19,36,33,45)(22,58,52,63)(23,43,25,61)(26,41,60,42)(40,53,49,62)(54,56,59,64)(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)(2,9)(4,22)(5,28)(6,17)(7,33)(8,20)(10,40)(11,45)(12,15)(13,48)(14,38)(16,25)(18,30)(19,29)(21,56)(23,55)(24,54)(26,62)(27,52)(31,34)(32,37)(35,42)(36,46)(39,59)(41,51)(43,58)(44,49)(47,50)(53,60)(57,64)(61,63);(1,4,5,23,29,60,31,39,8,27,30,25,7,26,6,24)(2,10,11,41,32,61,47,57,14,44,46,42,13,43,12,21)(3,16,17,52,33,59,18,53,20,55,34,22,19,54,28,62)(9,35,15,49,48,64,36,63,38,51,50,40,37,56,45,58)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6431);