Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C151 · Schur multiplier (H2) = C2 |
| direct product | C151×#13513 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C151 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C151 |
| socle | C2×C151 | #2085 |
| prime radical | C23×C151 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C151 ◁ C22×C4×C151 ◁ #405×C151 ◁ G | |
228 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,5,23,6,12,8,11)(3,14,16,9,18,34,30,35)(4,10,20,39,22,40,31,37)(7,26,24,41,25,13,29,57)(15,46,47,62,48,44,49,32)(17,42,27,61,51,45,58,33)(19,52,53,36,54,64,50,63)(21,55,28,60,56,38,59,43)(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);(1,3,5,16,6,18,8,30)(2,9,11,14,12,35,23,34)(4,19,20,53,22,54,31,50)(7,27,24,51,25,58,29,17)(10,36,37,52,40,63,39,64)(13,42,41,33,26,45,57,61)(15,28,47,56,48,59,49,21)(32,60,44,55,62,43,46,38);(1,4,5,20,6,22,8,31)(2,10,11,37,12,40,23,39)(3,15,16,47,18,48,30,49)(7,28,24,56,25,59,29,21)(9,32,14,44,35,62,34,46)(13,43,41,38,26,60,57,55)(17,50,27,19,51,53,58,54)(33,52,45,63,61,64,42,36)6040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 6501);