Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13078 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1893 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
454 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,6,13,8,12,7,31)(3,14,18,44,17,41,29,45)(4,19,21,40,22,10,30,39)(5,23,26,33,28,55,27,11)(9,32,35,16,37,51,36,24)(15,43,48,63,47,60,49,64)(20,53,54,42,25,56,57,38)(34,59,61,50,52,58,62,46)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191);(1,3,5,24,7,18,27,32,8,17,28,16,6,29,26,51)(2,9,33,14,13,36,55,45,12,37,11,41,31,35,23,44)(4,15,25,46,21,49,57,58,22,47,20,50,30,48,54,59)(10,34,38,43,40,61,56,63,19,52,42,60,39,62,53,64);(1,4)(2,10)(3,15)(5,25)(6,30)(7,21)(8,22)(9,34)(11,42)(12,19)(13,40)(14,43)(16,50)(17,47)(18,49)(20,28)(23,53)(24,46)(26,54)(27,57)(29,48)(31,39)(32,58)(33,38)(35,62)(36,61)(37,52)(41,60)(44,64)(45,63)(51,59)(55,56)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6066);