This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{5} \times C_{4}^{2}$ | |
| Order: | \(512\)\(\medspace = 2^{9} \) | |
| Exponent: | \(4\)\(\medspace = 2^{2} \) | |
| Automorphism group: | Group of order 1006568535490560 | |
| Nilpotency class: | $1$ | |
| Derived length: | $1$ |
This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary). Whether it is metacyclic or rational has not been computed.
Group statistics
| Order | 1 | 2 | 4 | ||
|---|---|---|---|---|---|
| Elements | 1 | 127 | 384 | 512 | |
| Conjugacy classes | 1 | 127 | 384 | 512 | |
| Divisions | data not computed | ||||
| Autjugacy classes | data not computed | ||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 512 | 512 |
Constructions
| Rank: | $7$ |
| Inequivalent generating 7-tuples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{5} \times C_{4}^{2}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{5} \times C_{4}^{2}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{5} \times C_{4}^{2}$ | |
| Frattini: | $\Phi \simeq$ $C_2^2$ | $G/\Phi \simeq$ $C_2^7$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{5} \times C_{4}^{2}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{5} \times C_{4}^{2}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^7$ | $G/S \simeq$ $C_2^2$ |