Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13656 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | C2×D16 |
| commutator | C16 | C22×C4×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C2×C127 | C4×D16 |
| prime radical | C127×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C2×C8×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,6,12,8,15,30,14)(3,16,19,47,21,48,20,54)(4,13,23,39,24,10,7,31)(5,25,27,57,29,38,62,58)(9,33,35,53,37,52,45,18)(11,22,41,55,43,56,42,28)(17,49,50,64,51,46,32,40)(26,44,59,63,61,34,60,36)(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)(2,9)(4,17)(5,26)(6,19)(7,32)(8,21)(10,34)(11,40)(12,35)(13,44)(14,45)(15,37)(16,25)(18,28)(20,30)(22,33)(23,50)(24,51)(27,59)(29,61)(31,36)(38,48)(39,63)(41,49)(42,46)(43,64)(47,57)(52,56)(53,55)(54,58)(60,62);(1,4,6,23,8,24,30,7)(2,10,12,31,15,13,14,39)(3,17,19,50,21,51,20,32)(5,22,27,55,29,56,62,28)(9,34,35,36,37,44,45,63)(11,38,41,58,43,25,42,57)(16,46,47,40,48,49,54,64)(18,26,33,59,53,61,52,60)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6644);