Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#13057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4×C127 |
| Frattini, cosocle | C22×Q8 | C23×C127 |
| prime radical | C2×#454×C127 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C23×C127 ◁ C22×Q8×C127 ◁ G | |
700 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 147 points:
(2,4)(5,6,7,11,10,15,8,13)(9,18,17,20,19,14,16,12)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,2,3,4)(6,11)(7,8)(12,14)(13,15)(16,17)(18,20);(5,7)(6,12)(8,10)(9,17)(11,18)(13,14)(15,20)(16,19)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6045);