Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C2×C4 |
| direct product | C127×#11771 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1505 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C2×M16 | C23×C127 |
| prime radical | C23×C127 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ #405×C127 ◁ G | |
292 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 292 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)(3,14,17,43)(4,10)(5,25,27,41)(6,12)(7,30)(8,13)(9,16)(11,28,39,29)(15,42,45,60)(18,38,19,49)(20,50,46,64)(21,52,53,62)(22,36)(23,56)(24,37)(26,40)(31,44)(32,47)(33,48)(34,51)(35,54,61,55)(57,59)(58,63)(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);(1,3,8,17)(2,9,13,33)(4,20,24,46)(5,26,27,47)(6,19,7,18)(10,34,37,59)(11,38,39,49)(12,40,30,32)(14,41,43,25)(15,23,45,22)(16,28,48,29)(21,44,53,63)(31,56,58,36)(35,60,61,42)(50,62,64,52)(51,54,57,55);(1,4,7,23,8,24,6,22)(2,10,30,56,13,37,12,36)(3,15,19,46,17,45,18,20)(5,21,29,55,27,53,28,54)(9,31,40,59,33,58,32,34)(11,35,41,62,39,61,25,52)(14,42,49,64,43,60,38,50)(16,44,26,57,48,63,47,51)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4759);