Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the Frattini subgroup, the Fitting subgroup, the radical, a direct factor, cyclic (hence elementary (for every $p$), hyperelementary, metacyclic, and a Z-group), stem, a $p$-group (for every $p$), perfect, and rational.
Ambient group ($G$) information
| Description: | $\SL(2,8)$ |
| Order: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Quotient group ($Q$) structure
| Description: | $\SL(2,8)$ |
| Order: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Automorphism Group: | ${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| Nilpotency class: | $-1$ |
| Derived length: | $0$ |
The quotient is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | ${}^2G(2,3)$, of order \(1512\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $\SL(2,8)$ | ||
| Normalizer: | $\SL(2,8)$ | ||
| Complements: | $\SL(2,8)$ | ||
| Minimal over-subgroups: | $C_7$ | $C_3$ | $C_2$ |
Other information
| Möbius function | $-504$ |
| Projective image | $\SL(2,8)$ |