Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C151 · Schur multiplier (H2) = C22 |
| direct product | C151×#13524 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C151 | #2092 |
| commutator, Frattini, cosocle | C22×C8 | C23×C151 |
| prime radical | C23×C151 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C151 ◁ C22×C4×C151 ◁ #405×C151 ◁ G | |
288 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 288 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:
(1,2,5,11,30,13,34,46,8,15,6,12,7,14,31,47)(3,16,18,50,19,48,32,38,21,52,57,37,20,44,28,9)(4,22,24,42,35,45,33,10,27,40,25,41,26,49,29,43)(17,36,53,58,23,59,60,64,55,62,61,63,54,39,56,51)(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);(1,3,7,19,8,21,30,20)(2,9,14,50,15,38,13,37)(4,23,26,55,27,54,35,17)(5,28,31,18,6,32,34,57)(10,39,42,36,43,59,41,62)(11,44,47,16,12,48,46,52)(22,58,49,64,40,63,45,51)(24,53,29,60,25,61,33,56);(1,4,8,27)(2,10,15,43)(3,17,21,55)(5,29,6,33)(7,35,30,26)(9,36,38,62)(11,45,12,49)(13,42,14,41)(16,51,52,64)(18,56,57,60)(19,54,20,23)(22,47,40,46)(24,34,25,31)(28,53,32,61)(37,59,50,39)(44,58,48,63)5134 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 6512);