Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#12026 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #427 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| socle | C2×C127 | #1568 |
| prime radical | C23×C127 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ #97×C127 ◁ #405×C127 ◁ G | |
180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 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:
(1,2,6,13,8,14,31,43,9,15,30,12,7,34,5,29)(3,16,19,36,21,45,18,37,22,41,32,58,20,54,52,10)(4,23,25,56,27,40,24,42,28,57,33,59,26,11,38,39)(17,46,48,35,50,60,62,55,51,64,53,61,49,44,47,63)(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);(1,3,5,18,7,20,30,19,9,22,31,52,8,21,6,32)(2,10,12,16,14,36,29,45,15,37,13,41,34,58,43,54)(4,17,24,47,26,49,25,53,28,51,38,62,27,50,33,48)(11,35,39,44,23,55,56,46,40,61,42,60,57,63,59,64);(1,4,7,26,9,28,8,27)(2,11,14,23,15,40,34,57)(3,17,20,49,22,51,21,50)(5,24,30,25,31,38,6,33)(10,35,36,55,37,61,58,63)(12,39,29,56,13,42,43,59)(16,44,45,46,41,60,54,64)(18,47,19,53,52,62,32,48)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5014);