Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C2×C127×#1342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #327 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C2×M16 | C23×C127 |
| prime radical | C24×C127 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×M16×C127 ◁ G | |
428 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);(3,4,7,12,9,13,8,24)(5,14,16,28,18,31,17,34)(6,19,21,11,23,29,22,30)(10,20,25,33,27,32,26,15);(3,5,9,18)(4,10,13,27)(6,20,23,32)(7,17,8,16)(11,28,30,34)(12,25,24,26)(14,29,31,19)(15,22,33,21)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4675);