Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13489 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #2073 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
278 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 278 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)(3,15,19,9)(4,20,24,44)(5,12)(6,13)(7,14)(8,11)(10,22,36,31)(16,47)(17,43,30,33)(18,46,52,34)(21,54)(23,41,57,64)(25,39)(26,58,59,40)(27,60,61,37)(28,42)(29,38)(32,48)(35,55)(45,49)(50,62)(51,63)(53,56)(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)(4,21)(5,17)(6,26)(7,30)(8,19)(10,35)(11,15)(12,33)(13,40)(14,43)(16,23)(18,28)(20,53)(22,55)(24,56)(25,52)(27,48)(29,59)(31,51)(32,37)(34,42)(36,63)(38,58)(39,46)(41,62)(44,54)(45,60)(47,64)(49,61)(50,57);(1,4)(2,10,11,36)(3,16)(5,22)(6,27)(7,31)(8,24)(9,32,15,45)(12,20,14,44)(13,41,38,64)(17,48)(18,51)(19,50)(21,26)(23,28)(25,57)(29,61)(30,49)(33,62,43,47)(34,54,46,53)(35,58,63,40)(37,39,60,42)(52,55)(56,59)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6477);