Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#8528 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #379 |
| commutator | C42 | C22×C4×C127 |
| Frattini, cosocle | C2×C42 | C23×C127 |
| prime radical | C127×#3545 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 167 points:
(2,6)(3,4)(7,8)(9,10)(11,19,22,15)(12,23,24,34)(13,18)(14,17)(16,25,31,27)(20,36)(21,33,26,30)(28,32)(29,38)(35,39)(37,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,29)(17,19)(18,33)(21,28)(23,36)(24,37)(25,39)(27,38)(30,32)(31,35)(34,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,20)(13,27)(14,24)(15,29)(17,31)(18,34)(19,35)(21,39)(22,37)(23,32)(25,28)(26,38)(30,36)(33,40)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 1516);