Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#11332 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #121 |
| commutator | C22×C4 | C22×C4×C127 |
| Frattini, cosocle | C23×C4 | C23×C127 |
| socle | C22×C127 | #389 |
| prime radical | C24×C127 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(2,4)(5,6,10,14,11,15,9,26)(7,16,19,28,20,33,18,12)(8,21,23,31,25,30,24,13)(17,34,22,29,36,27,35,32)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,2,3,4)(5,7,9,18,11,20,10,19)(6,12,14,16,15,28,26,33)(8,22,23,36,25,35,24,17)(13,29,30,34,31,32,21,27);(5,8,9,24,11,25,10,23)(6,13,14,21,15,31,26,30)(7,17,18,35,20,36,19,22)(12,27,16,32,28,34,33,29)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4320);