Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13638 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #367 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C2×C127 | C2×#367 |
| prime radical | C23×C127 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G | |
182 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)(3,14,18,41)(4,9)(5,23)(6,27)(7,13)(8,17,31,51)(10,25)(11,21)(12,22)(15,34)(16,39,48,43)(19,54,53,61)(20,55)(24,38)(26,57)(28,35)(29,56)(30,47,33,60)(32,44,59,64)(36,46)(37,45)(40,62,58,52)(42,50,63,49)(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);(1,3,15,44,6,19,5,17,22,48,25,52,24,50,56,60,7,18,46,64,28,53,26,51,4,16,45,62,21,49,20,47)(2,8,29,54,11,32,10,14,9,30,23,42,35,40,34,39,13,31,55,61,38,59,37,41,12,33,57,63,27,58,36,43);(1,4,7,22)(2,9,13,12)(3,16,18,48)(5,20,26,56)(6,21,28,24)(8,30,31,33)(10,34,37,36)(11,35,38,27)(14,39,41,43)(15,45,46,25)(17,47,51,60)(19,49,53,50)(23,55,57,29)(32,40,59,58)(42,61,63,54)(44,62,64,52)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6626);