Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C2×C4 |
| direct product | C127×#13146 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #466 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1934 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,6,12,8,13,7,30)(3,14,17,42,19,43,18,52)(4,20,23,55,25,37,24,57)(5,26,27,32,29,58,28,11)(9,31,34,51,35,50,41,16)(10,36,38,21,39,56,54,22)(15,44,47,60,49,62,48,33)(40,61,63,45,53,59,64,46)(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);(1,3,5,16,6,17,27,31,8,19,29,51,7,18,28,50)(2,9,32,43,30,41,26,42,13,35,11,14,12,34,58,52)(4,15,21,45,23,47,56,59,25,49,22,46,24,48,36,61)(10,33,20,53,54,62,57,63,39,60,37,40,38,44,55,64);(1,4,5,21,6,23,27,56,8,25,29,22,7,24,28,36)(2,10,11,37,30,54,58,55,13,39,32,20,12,38,26,57)(3,15,16,45,17,47,31,59,19,49,51,46,18,48,50,61)(9,33,14,40,41,62,52,64,35,60,43,53,34,44,42,63)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6134);