Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C151 · Schur multiplier (H2) = C22 |
| direct product | C151×#11990 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C151 | #1551 |
| commutator | C42 | C22×C4×C151 |
| Frattini, cosocle | C4×C8 | C23×C151 |
| prime radical | C23×C151 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C151 ◁ C2×C4×C151 ◁ #405×C151 ◁ G | |
268 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 268 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) = 215 |
| transitive degree | μt(G) = 9664 |
A minimal faithful permutation representation — 3 generators on 215 points:
(1,2,5,11,6,10,24,36,7,13,25,39,27,12,40,38)(3,14,16,42,17,31,46,56,18,30,47,37,26,8,23,29)(4,19,21,48,22,9,32,33,20,35,49,61,50,34,58,60)(15,28,44,54,45,41,62,63,43,55,57,64,52,53,51,59)(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);(1,3,7,18)(2,8,13,31)(4,15,20,43)(5,23,25,46)(6,26,27,17)(9,28,34,55)(10,30,12,14)(11,37,39,42)(16,24,47,40)(19,41,35,53)(21,51,49,62)(22,52,50,45)(29,36,56,38)(32,57,58,44)(33,59,60,63)(48,54,61,64);(1,4)(2,9,13,34)(3,15)(5,21)(6,22)(7,20)(8,28,31,55)(10,35,12,19)(11,33,39,60)(14,41,30,53)(16,44)(17,45)(18,43)(23,51)(24,32)(25,49)(26,52)(27,50)(29,54,56,64)(36,61,38,48)(37,59,42,63)(40,58)(46,62)(47,57)5134 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 4978);