Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C151 · Schur multiplier (H2) = C2 |
| direct product | C151×#13520 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C151 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C151 |
| socle | C2×C151 | #2087 |
| prime radical | C23×C151 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C151 ◁ C22×C4×C151 ◁ #405×C151 ◁ G | |
238 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 215 |
| transitive degree | μt(G) = 9664 |
A minimal faithful permutation representation — 3 generators on 215 points:
(1,2,5,10,23,34,52,61,7,11,25,35,6,12,24,36)(3,8,14,28,42,57,62,64,15,30,43,59,16,29,44,58)(4,17,20,37,26,53,51,31,19,46,49,32,21,33,22,9)(13,38,39,60,45,63,50,54,18,47,48,55,40,56,41,27)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215);(1,3)(2,8)(4,18,19,13)(5,14)(6,16)(7,15)(9,27,31,54)(10,28)(11,30)(12,29)(17,38,46,47)(20,48,49,39)(21,45,26,40)(22,50,51,41)(23,42)(24,44)(25,43)(32,55,37,60)(33,56,53,63)(34,57)(35,59)(36,58)(52,62)(61,64);(1,4,6,26,7,19,23,21)(2,9,12,37,11,31,34,32)(3,13,16,45,15,18,42,40)(5,22,24,20,25,51,52,49)(8,27,29,60,30,54,57,55)(10,33,36,17,35,53,61,46)(14,41,44,39,43,50,62,48)(28,56,58,38,59,63,64,47)6040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 6508);