Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#11871 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1519 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C2×M16 | C23×C127 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×#131×C127 ◁ G | |
360 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 360 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) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,6,12,8,13,7,32)(3,14,17,40,19,45,18,53)(4,20,23,39,25,55,24,10)(5,26,29,11,31,42,30,43)(9,27,34,52,36,51,35,16)(15,46,21,56,49,63,48,44)(22,41,58,54,28,38,59,60)(33,47,61,64,62,50,37,57)(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);(1,3,8,19)(2,9,13,36)(4,21,25,48)(5,27,31,51)(6,18,7,17)(10,37,39,61)(11,40,43,53)(12,34,32,35)(14,42,45,26)(15,24,49,23)(16,30,52,29)(20,33,55,62)(22,47,28,50)(38,44,41,56)(46,54,63,60)(57,59,64,58);(1,4,7,24,8,25,6,23)(2,10,12,20,13,39,32,55)(3,15,18,48,19,49,17,21)(5,28,29,59,31,22,30,58)(9,33,35,37,36,62,34,61)(11,41,26,60,43,38,42,54)(14,44,40,46,45,56,53,63)(16,50,27,57,52,47,51,64)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4859);