Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C25 |
| direct product | C2×C127×#1664 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C127 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C127 |
| prime radical | C22×D4×C127 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G | |
894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 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);(3,4,8,12)(5,13,15,18)(6,11,7,10)(9,16,17,14)(19,20,24,22)(21,27,30,34)(23,31,33,26)(25,32,28,29);(3,5)(4,9)(6,16)(7,14)(8,15)(10,18)(11,13)(12,17)(19,21,24,30)(20,25,22,28)(23,32,33,29)(26,34,31,27);(3,5)(4,9)(6,14)(7,16)(8,15)(10,13)(11,18)(12,17)(19,22,24,20)(21,28,30,25)(23,26,33,31)(27,32,34,29)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11974);