Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group (for every $p$), perfect, and rational.
Ambient group ($G$) information
| Description: | $D_4^2:C_2^3$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
Quotient group ($Q$) structure
| Description: | $D_4^2:C_2^3$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^5.D_4^3$, of order \(16384\)\(\medspace = 2^{14} \) |
| Outer Automorphisms: | $D_4^2$, of order \(64\)\(\medspace = 2^{6} \) |
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.D_4^3$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\card{W}$ | $1$ |
Related subgroups
| Centralizer: | $D_4^2:C_2^3$ | ||||||||||
| Normalizer: | $D_4^2:C_2^3$ | ||||||||||
| Complements: | $D_4^2:C_2^3$ | ||||||||||
| Minimal over-subgroups: | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | not computed |