Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13503 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #2077 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 272 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) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,8,30,27,11,6,12,7,14,32,40,5,13,29,42)(3,15,31,61,17,9,18,45,20,39,49,34,19,35,51,64)(4,21,28,38,23,57,24,37,25,54,58,10,26,41,59,43)(16,46,50,63,52,53,48,33,22,55,60,44,47,62,56,36)(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)(4,22)(5,19)(6,18)(7,20)(8,31)(10,36)(11,39)(12,34)(13,15)(14,35)(16,25)(17,27)(21,53)(23,47)(24,56)(26,52)(28,60)(29,51)(30,45)(32,49)(33,38)(37,44)(40,64)(41,46)(42,61)(43,63)(48,59)(50,58)(54,62)(55,57);(1,4,5,23,7,25,27,26)(2,10,11,37,14,38,13,43)(3,16,17,47,20,22,19,52)(6,28,8,24,29,58,32,59)(9,33,15,44,35,36,39,63)(12,41,40,54,42,57,30,21)(18,50,49,56,51,60,31,48)(34,55,45,62,61,46,64,53)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6491);