Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#7099 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #106 |
| commutator | C8 | C2×C16×C127 |
| Frattini, cosocle | C82 | C22×C127 |
| socle, prime radical | C22×C127 | #334 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ C82×C127 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 2 generators on 159 points:
(2,6,7,12)(3,9)(4,11)(8,15,14,13)(10,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(1,2,3,6,4,8,9,13,5,7,10,12,11,14,16,15)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 87);