Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13479 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C23×C127 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×C8×C127 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2,3,4)(5,6)(7,10)(8,18)(9,14)(11,19)(12,17)(13,22)(15,27)(16,25)(20,29)(21,32)(23,30)(24,34)(26,33)(28,31)(35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(5,7,10,6,11,14,26,16,12,15,27,17,25,33,9,19)(8,20,22,34,23,35,36,30,24,13,29,18,28,32,21,31);(2,4)(5,8,12,24)(6,13,17,20)(7,18,15,34)(9,22,26,29)(10,21,27,36)(11,28,25,23)(14,30,33,31)(16,35,19,32)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6467);