Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#17897 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C22×D4 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C42 | C24×C127 |
| socle | C22×C127 | C2×C4×D4 |
| prime radical | C23×C127 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ #431×C127 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 167 points:
(1,2,5,3,4,6,8,7)(9,10)(11,19)(12,16)(13,25)(14,17)(15,21)(18,27)(20,33)(22,32)(23,31)(24,37)(26,29)(28,35)(30,36)(34,40)(38,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(2,6)(3,7)(9,11,14,22)(10,15,17,29)(12,20,23,34)(13,26,27,21)(16,28,31,39)(18,32,25,19)(24,38,36,35)(30,40,37,33);(1,3,5,6,4,7,8,2)(9,12,14,23)(10,16,17,31)(11,20,22,34)(13,24,27,36)(15,28,29,39)(18,30,25,37)(19,33,32,40)(21,35,26,38);(10,17)(12,23)(15,29)(18,25)(19,32)(20,34)(24,36)(35,38)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 10885);