Properties

Label 32T12882
Degree $32$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes

Related objects

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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 384

Siblings 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.