Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C127 · Schur multiplier (H2) = C2×C4 |
| direct product | C127×#7190 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #99 |
| commutator | C2×C4 | C4×C8×C127 |
| Frattini, cosocle | C2×#117 | C22×C127 |
| socle | C22×C127 | #342 |
| prime radical | C24×C127 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×#117×C127 ◁ G | |
322 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 322 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) = 167 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 2 generators on 167 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,21,31)(12,23,24,16)(13,26)(14,18)(15,20)(17,27)(22,36,35,32)(25,30,38,37)(28,40)(29,34)(33,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(9,11,13,22,14,21,27,35)(10,15,17,28,18,29,26,39)(12,20,25,33,24,34,38,40)(16,19,30,32,23,31,37,36)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 178);