Group action invariants
| Degree $n$: | $32$ | |
| Transitive number $t$: | $12882$ | |
| Parity: | $1$ | |
| Primitive: | no | |
| Nilpotency class: | $4$ | |
| $|\Aut(F/K)|$: | $8$ | |
| Generators: | (1,28,17,10)(2,27,18,9)(3,26,19,12)(4,25,20,11)(5,6)(7,8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32), (1,21,27,29,17,7,9,15)(2,22,28,30,18,8,10,16)(3,23,25,31,19,5,11,13)(4,24,26,32,20,6,12,14), (1,14,9,24,17,32,27,6)(2,13,10,23,18,31,28,5)(3,16,11,22,19,30,25,8)(4,15,12,21,20,29,26,7), (1,32)(2,31)(3,30)(4,29)(5,10,23,28)(6,9,24,27)(7,12,21,26)(8,11,22,25)(13,18)(14,17)(15,20)(16,19), (1,16,27,8,17,30,9,22)(2,15,28,7,18,29,10,21)(3,14,25,6,19,32,11,24)(4,13,26,5,20,31,12,23) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 31 $4$: $C_2^2$ x 155 $8$: $D_{4}$ x 24, $C_2^3$ x 155 $16$: $D_4\times C_2$ x 84, $C_2^4$ x 31 $32$: $C_2^2 \wr C_2$ x 16, $C_2^2 \times D_4$ x 42, 32T39 $64$: $(((C_4 \times C_2): C_2):C_2):C_2$ x 4, 16T105 x 12, 32T273 x 3 $128$: $C_2 \wr C_2\wr C_2$ x 4, 16T245 x 6, 32T1369 $256$: 16T509 x 6, 32T4287 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 7
Degree 4: $C_2^2$ x 7, $D_{4}$ x 4
Degree 8: $C_2^3$, $D_4\times C_2$ x 6, $C_2 \wr C_2\wr C_2$ x 4
Degree 16: $C_2^2 \times D_4$, 16T509 x 6
Low degree siblings
32T12882 x 127, 32T12885 x 384Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 80 conjugacy classes of elements. Data not shown.
Group invariants
| Order: | $512=2^{9}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | not available |
| Character table: not available. |