Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13164 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | #121 |
| commutator | C2×C4 | C22×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1939 |
| prime radical | C23×C127 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G | |
284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,6,12,7,11,22,29,8,13,21,10,5,23,4,20)(3,14,18,31,16,9,24,27,19,26,32,30,17,28,15,25)(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);(1,3,6,18,7,16,22,24,8,19,21,32,5,17,4,15)(2,9,12,27,11,26,29,30,13,28,10,25,23,14,20,31);(1,4,5,21,8,22,7,6)(2,10,11,20,13,12,23,29)(3,15,17,32,19,24,16,18)(9,25,26,31,28,27,14,30)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6152);