Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13612 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2127 |
| commutator | C42 | C22×C4×C127 |
| Frattini, cosocle | C4⋊Q8 | C23×C127 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C127 ◁ C22×C4×C127 ◁ #536×C127 ◁ G | |
304 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 304 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:
(1,2,5,11,7,13,25,40)(3,14,16,38,18,31,45,57)(4,9,21,34,20,35,49,62)(6,12,27,39,28,10,24,37)(8,23,30,26,32,44,58,17)(15,41,42,60,19,46,47,63)(22,33,53,59,50,36,52,61)(29,54,55,48,56,51,64,43)(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);(1,3,7,18)(2,8,13,32)(4,19,20,15)(5,23,25,44)(6,26,28,17)(9,29,35,56)(10,31,12,14)(11,38,40,57)(16,24,45,27)(21,51,49,54)(22,43,50,48)(30,39,58,37)(33,41,36,46)(34,60,62,63)(42,53,47,52)(55,59,64,61);(1,4)(2,9)(3,15)(5,21)(6,22)(7,20)(8,29)(10,36)(11,34)(12,33)(13,35)(14,41)(16,42)(17,43)(18,19)(23,54)(24,52)(25,49)(26,48)(27,53)(28,50)(30,55)(31,46)(32,56)(37,61)(38,60)(39,59)(40,62)(44,51)(45,47)(57,63)(58,64)3175 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6600);