Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13126 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | #124 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1924 |
| prime radical | C2×#41×C127 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,6,12,7,13,30,45,8,14,31,46,5,11,16,33)(3,15,20,47,18,37,48,43,21,35,28,59,19,9,32,36)(4,22,26,55,25,42,49,44,27,40,29,60,24,10,38,41)(17,50,51,62,23,56,57,64,52,34,54,61,53,39,58,63)(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);(1,3,16,48,5,19,31,20,8,21,30,32,7,18,6,28)(2,9,33,47,11,35,46,36,14,37,45,59,13,15,12,43)(4,23,38,54,24,17,29,57,27,53,49,51,25,52,26,58)(10,39,60,62,40,34,44,63,42,56,55,61,22,50,41,64);(1,4,5,24,8,27,7,25)(2,10,11,40,14,42,13,22)(3,17,18,23,21,52,19,53)(6,29,16,49,31,26,30,38)(9,34,15,39,37,50,35,56)(12,44,33,55,46,41,45,60)(20,54,48,58,28,51,32,57)(36,62,47,64,43,61,59,63)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6114);