Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13702 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C23×C127 |
| prime radical | C127×#2221 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ #131×C127 ◁ #504×C127 ◁ G | |
184 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(2,9)(3,16)(4,20)(5,7)(6,21)(10,24)(11,38)(12,32)(13,44)(14,30)(15,33)(17,48)(18,51)(19,25)(22,36)(23,37)(26,27)(28,39)(29,31)(34,43)(35,52)(40,59)(41,57)(42,56)(45,58)(46,49)(50,60)(54,62)(55,61)(63,64)(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);(1,2,10,34,6,13,23,32,5,12,36,44,26,43,4,9,8,15,20,31,27,45,22,30,7,14,37,58,21,29,24,33)(3,11,35,59,18,40,46,38,25,42,48,62,53,63,17,28,19,41,49,55,51,61,52,57,16,39,60,64,47,54,50,56);(1,3)(2,11)(4,17)(5,25)(6,18)(7,16)(8,19)(9,28)(10,35)(12,42)(13,40)(14,39)(15,41)(20,49)(21,47)(22,52)(23,46)(24,50)(26,53)(27,51)(29,54)(30,57)(31,55)(32,38)(33,56)(34,59)(36,48)(37,60)(43,63)(44,62)(45,61)(58,64)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6690);