Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C127 · Schur multiplier (H2) = C22×C4 |
| direct product | C127×#12592 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C127 | C23 |
| commutator | C22 | C2×C4×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle, prime radical | C23×C127 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C127 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 171 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 3 generators on 171 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,20,17,21,26,32,18,22,27,33,28,34,38,42)(15,19,24,30,25,31,37,41,23,29,35,39,36,40,43,44)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(1,2,3,4)(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,30)(21,31)(22,29)(26,37)(27,35)(28,36)(32,41)(33,39)(34,40)(38,43)(42,44);(5,7,8,11)(6,10,9,12)(15,23)(19,29)(24,35)(25,36)(30,39)(31,40)(37,43)(41,44)14224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5580);