Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C28 |
| direct product | C2×C127×#1796 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #379 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C22×2−1+4×C127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G | |
3602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 8128 |
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)(5,13)(6,17,19,10)(7,20)(8,12)(9,15)(11,22)(14,28,30,27)(16,26)(18,25,34,32)(21,24)(23,29,31,33);(3,5,8,16)(4,9,12,24)(6,14,19,30)(7,21,22,15)(10,23,17,31)(11,26,20,13)(18,33,34,29)(25,28,32,27);(3,6)(4,10)(5,14)(7,18)(8,19)(9,23)(11,25)(12,17)(13,27)(15,29)(16,30)(20,32)(21,33)(22,34)(24,31)(26,28)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13610);