Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13447 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #2054 |
| prime radical | C127×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
386 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 386 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) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2)(3,16)(4,22)(5,29)(6,32)(7,12)(8,34)(9,15)(10,18)(11,25)(13,31)(14,30)(17,36)(19,46)(20,38)(21,45)(23,47)(24,55)(26,44)(27,58)(28,52)(33,40)(35,37)(39,51)(41,53)(42,56)(43,57)(48,60)(49,63)(50,62)(54,64)(59,61)(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);(1,3)(2,10)(4,23)(5,19)(6,33)(7,18)(8,35)(9,21)(11,41)(12,37)(13,45)(14,46)(15,40)(16,29)(17,26)(20,30)(22,36)(24,49)(25,50)(27,48)(28,54)(31,51)(32,38)(34,39)(42,63)(43,64)(44,60)(47,55)(52,62)(53,56)(57,59)(58,61);(1,4,5,24,30,56,6,25,9,28,31,57,8,27,7,26)(2,11,12,42,34,55,13,22,15,44,32,58,14,43,29,52)(3,17,18,48,35,59,51,54,21,50,33,53,20,49,19,23)(10,36,16,47,46,63,38,41,40,62,45,64,39,61,37,60)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6435);