This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_2^2 . C_2^7$ | |
| Order: | \(512\)\(\medspace = 2^{9} \) | |
| Exponent: | \(4\)\(\medspace = 2^{2} \) | |
| Automorphism group: | Group of order 2097152 | |
| Nilpotency class: | $2$ | |
| Derived length: | $2$ |
This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. Whether it is metacyclic, monomial, or rational has not been computed.
Group statistics
| Order | 1 | 2 | 4 | ||
|---|---|---|---|---|---|
| Elements | 1 | 207 | 304 | 512 | |
| Conjugacy classes | 1 | 65 | 80 | 146 | |
| Divisions | data not computed | ||||
| Autjugacy classes | data not computed | ||||
| Dimension | 1 | 4 | 8 | |
|---|---|---|---|---|
| Irr. complex chars. | 128 | 16 | 2 | 146 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
| |||||
Homology
| Abelianization: | $C_{2}^{7} $ |
Subgroups
| Center: | $Z \simeq$ $C_2^2$ | $G/Z \simeq$ $C_2^7$ | |
| Commutator: | $G' \simeq$ $C_2^2$ | $G/G' \simeq$ $C_2^7$ | |
| Frattini: | $\Phi \simeq$ $C_2^2$ | $G/\Phi \simeq$ $C_2^7$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_2^2 . C_2^7$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_2^2 . C_2^7$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^2$ | $G/S \simeq$ $C_2^7$ | |
| Maximal subgroups: | $M_{2,1} \simeq$ $C_2^2\times D_4^2$ | $G/M_{2,1} \simeq$ $C_2$ | |
| $M_{2,2} \simeq$ $C_4^2:C_2^4$ | $G/M_{2,2} \simeq$ $C_2$ | 8 normal subgroups | |
| $M_{2,3} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,3} \simeq$ $C_2$ | 4 normal subgroups | |
| $M_{2,4} \simeq$ $C_4^2:C_2^4$ | $G/M_{2,4} \simeq$ $C_2$ | 8 normal subgroups | |
| $M_{2,5} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,5} \simeq$ $C_2$ | 8 normal subgroups | |
| $M_{2,6} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,6} \simeq$ $C_2$ | 2 normal subgroups | |
| $M_{2,7} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,7} \simeq$ $C_2$ | 32 normal subgroups | |
| $M_{2,8} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,8} \simeq$ $C_2$ | 16 normal subgroups | |
| $M_{2,9} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,9} \simeq$ $C_2$ | 8 normal subgroups | |
| $M_{2,10} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,10} \simeq$ $C_2$ | 8 normal subgroups | |
| $M_{2,11} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,11} \simeq$ $C_2$ | 16 normal subgroups | |
| $M_{2,12} \simeq$ $D_4^2:C_2^2$ | $G/M_{2,12} \simeq$ $C_2$ | 16 normal subgroups | |
| Maximal quotients: | $m_{2,1} \simeq$ $C_2$ | $G/m_{2,1} \simeq$ $D_4:C_2^5$ | 2 normal subgroups |
| $m_{2,2} \simeq$ $C_2$ | $G/m_{2,2} \simeq$ $C_2^4:C_2^4$ |