Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12878 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #418 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1824 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
362 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2,6,13,8,11,7,30)(3,14,17,35,18,9,29,34)(4,19,22,51,20,52,31,41)(5,23,26,39,28,55,27,12)(10,25,36,53,38,59,37,21)(15,43,46,64,44,60,47,62)(16,48,50,40,24,56,58,33)(32,49,61,45,42,57,63,54)(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)(2,9)(4,15)(5,24)(6,29)(7,17)(8,18)(10,32)(11,14)(12,40)(13,35)(16,28)(19,43)(20,44)(21,54)(22,47)(23,48)(25,57)(26,50)(27,58)(30,34)(31,46)(33,39)(36,63)(37,61)(38,42)(41,64)(45,53)(49,59)(51,62)(52,60)(55,56);(1,4,7,31,8,20,6,22)(2,10,30,37,11,38,13,36)(3,15,17,46,18,44,29,47)(5,25,27,21,28,59,26,53)(9,32,34,61,14,42,35,63)(12,41,55,52,39,51,23,19)(16,49,50,45,24,57,58,54)(33,62,48,43,40,64,56,60)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5866);