Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C24 |
| direct product | C2×C127×#1392 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #99 |
| commutator | C23 | C2×C42×C127 |
| Frattini, cosocle | C2×#41 | C23×C127 |
| socle | C22×C127 | #342 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 161 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161);(3,4,7,12,9,14,8,13)(5,15,17,30,19,32,18,31)(6,20,21,29,22,11,23,27)(10,24,25,16,26,33,28,34);(3,5,8,18,9,19,7,17)(4,10,13,25,14,26,12,28)(6,16,21,33,22,34,23,24)(11,15,27,31,20,32,29,30)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5566);