Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13500 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #2076 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,6,11,26,40,57,53,7,12,29,43,5,10,25,39)(3,13,16,45,15,34,50,41,18,33,27,61,17,8,30,32)(4,19,22,38,24,59,60,37,23,54,58,36,21,42,28,9)(14,46,48,31,47,52,62,63,20,55,56,35,49,64,51,44)(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)(2,8)(4,20)(5,17)(6,27)(7,18)(9,35)(10,33)(11,41)(12,34)(13,40)(14,23)(15,26)(16,29)(19,52)(21,47)(22,48)(24,49)(25,50)(28,51)(30,57)(31,37)(32,43)(36,63)(38,44)(39,45)(42,46)(53,61)(54,64)(55,59)(56,58)(60,62);(1,4,7,23)(2,9,12,37)(3,14)(5,24,26,21)(6,28,29,60)(8,31)(10,38,40,36)(11,42,43,59)(13,44)(15,49)(16,51)(17,47)(18,20)(19,53,54,39)(22,57,58,25)(27,62)(30,48)(32,46)(33,63)(34,35)(41,55)(45,64)(50,56)(52,61)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6488);