Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C2×C127×#1868 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #120 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C22×C127 | #385 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G | |
1342 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);(3,4,7,11,8,12,18,24)(5,9,15,21,14,20,28,32)(6,10,17,23,16,22,29,33)(13,25,27,34,26,19,30,31);(3,5,7,15,8,14,18,28)(4,9,11,21,12,20,24,32)(6,13,17,27,16,26,29,30)(10,19,23,31,22,25,33,34);(3,6)(4,10)(5,13)(7,17)(8,16)(9,19)(11,23)(12,22)(14,26)(15,27)(18,29)(20,25)(21,31)(24,33)(28,30)(32,34)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13879);