Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13173 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #466 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1942 |
| prime radical | C23×C127 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,5,13,8,14,7,11)(3,15,19,32,20,37,17,9)(4,21,23,53,26,54,25,52)(6,28,27,39,29,38,22,12)(10,33,35,56,36,55,34,24)(16,42,46,61,47,62,44,51)(18,48,49,60,50,31,43,41)(30,45,40,58,57,64,59,63)(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);(1,3,5,19,8,20,7,17)(2,9,13,15,14,32,11,37)(4,16,23,46,26,47,25,44)(6,18,27,49,29,50,22,43)(10,30,35,40,36,57,34,59)(12,31,28,41,39,48,38,60)(21,51,53,42,54,61,52,62)(24,45,33,58,56,64,55,63);(1,4,22,55,7,25,29,56,8,26,27,33,5,23,6,24)(2,10,28,53,13,35,39,54,14,36,38,52,11,34,12,21)(3,16,43,63,17,44,50,64,20,47,49,58,19,46,18,45)(9,30,41,42,15,40,48,61,32,57,60,62,37,59,31,51)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6161);