Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C27 |
| direct product | C127×#20986 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C24 | C24×C127 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C23×Q8×C127 ◁ G | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2)(3,4)(5,6)(7,11,17,24)(8,12)(9,13)(10,14)(15,22,30,34)(16,23,32,36)(18,25)(19,26)(20,27)(21,28)(29,33,31,35)(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)(6,11,14,24)(8,15)(9,16)(10,17)(12,22,26,34)(13,23,28,36)(18,29)(19,30)(20,31)(21,32)(25,33,27,35);(2,4)(5,8,10,19)(6,12,14,26)(7,15,17,30)(9,20,21,18)(11,22,24,34)(13,27,28,25)(16,31,32,29)(23,35,36,33);(5,9,10,21)(6,13,14,28)(7,16,17,32)(8,18,19,20)(11,23,24,36)(12,25,26,27)(15,29,30,31)(22,33,34,35)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 13974);