Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13441 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C23×C127 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
378 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 195 points:
(1,2)(3,4)(5,6,11,16)(7,17,21,49)(8,22,27,40)(9,28,32,47)(10,33,30,15)(12,35,38,19)(13,24,42,25)(14,43,45,31)(18,50,53,66)(20,54,34,44)(23,48,51,39)(26,62,59,56)(29,46,64,37)(36,57,65,52)(41,61,55,60)(58,67,63,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(5,7,11,21)(6,12,16,38)(8,23,27,51)(9,29,32,64)(10,34,30,20)(13,39,42,48)(14,44,45,54)(15,46,33,37)(17,47,49,28)(18,24,53,25)(19,43,35,31)(22,36,40,65)(26,63,59,58)(41,68,55,67)(50,56,66,62)(52,61,57,60);(2,4)(5,8,9,24,30,59,43,61,11,27,32,25,10,26,31,60)(6,13,14,40,33,55,47,56,16,42,45,22,15,41,28,62)(7,18,19,51,34,57,64,58,21,53,35,23,20,52,29,63)(12,36,17,48,46,66,54,67,38,65,49,39,37,50,44,68)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6429);