Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C151 · Schur multiplier (H2) = C22 |
| direct product | C151×#13742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C151 | C2×D32 |
| commutator, Frattini, cosocle | C32 | C23×C151 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C151 ◁ C2×C4×C151 ◁ C2×C8×C151 ◁ C2×C16×C151 ◁ C2×C32×C151 ◁ G | |
134 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,8)(4,19)(5,23)(6,26)(7,9)(10,14)(11,15)(12,20)(13,24)(16,37)(17,38)(18,27)(21,54)(22,50)(25,56)(28,40)(29,39)(30,47)(31,42)(32,45)(33,53)(34,46)(35,49)(36,55)(41,57)(43,52)(44,51)(48,61)(58,63)(59,62)(60,64)(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);(2,8)(3,14)(4,15)(5,20)(6,24)(9,27)(10,11)(12,37)(13,38)(16,46)(17,49)(18,42)(19,28)(21,55)(22,45)(23,29)(25,53)(26,30)(31,32)(33,61)(34,52)(35,51)(36,57)(39,40)(41,47)(43,44)(48,64)(50,58)(54,60)(56,59)(62,63);(2,9)(8,27)(10,31)(11,32)(12,33)(13,26)(19,50)(23,56)(28,58)(29,59)(30,38)(34,60)(35,57)(36,51)(37,61)(52,54)6946 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 6730);