Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#13544 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #2101 |
| prime radical | C23×C127 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
212 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,15,18,43)(4,10,22,37)(5,11)(6,14)(7,12)(8,13)(9,17,33,29)(16,45)(19,39,50,63)(20,52)(21,35,30,44)(23,36,55,62)(24,38)(25,42)(26,57,56,32)(27,40,46,64)(28,41)(31,49)(34,51)(47,59)(48,58)(53,61)(54,60)(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);(1,3,8,29,6,18,5,17)(2,9,11,15,14,33,13,43)(4,20,30,51,22,54,21,53)(7,26,28,19,25,56,24,50)(10,34,35,52,37,61,44,60)(12,39,38,57,42,63,41,32)(16,46,49,55,48,27,47,23)(31,40,45,62,59,64,58,36);(1,4,5,21,6,22,8,30)(2,10,11,35,14,37,13,44)(3,16,17,47,18,48,29,49)(7,27,24,55,25,46,28,23)(9,31,15,45,33,59,43,58)(12,40,38,62,42,64,41,36)(19,51,26,20,50,53,56,54)(32,60,39,34,57,52,63,61)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6532);