This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_2^4 . C_2^5$ | |
| Order: | \(512\)\(\medspace = 2^{9} \) | |
| Exponent: | \(4\)\(\medspace = 2^{2} \) | |
| Automorphism group: | Group of order 335544320 | |
| 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 | 191 | 320 | 512 | |
| Conjugacy classes | 1 | 51 | 40 | 92 | |
| Divisions | data not computed | ||||
| Autjugacy classes | data not computed | ||||
| Dimension | 1 | 2 | 4 | |
|---|---|---|---|---|
| Irr. complex chars. | 32 | 40 | 20 | 92 |
Constructions
| Presentation: |
${\langle a, b, c, d, e, f, g, h, i \mid c^{2}=d^{2}=e^{2}=f^{2}=g^{2}=h^{2}= \!\cdots\! \rangle}$
| |||||
Homology
| Abelianization: | $C_{2}^{5} $ |
Subgroups
| Center: | $Z \simeq$ $C_2^5$ | $G/Z \simeq$ $C_2^4$ | |
| Commutator: | $G' \simeq$ $C_2^4$ | $G/G' \simeq$ $C_2^5$ | |
| Frattini: | $\Phi \simeq$ $C_2^4$ | $G/\Phi \simeq$ $C_2^5$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_2^4 . C_2^5$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_2^4 . C_2^5$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^5$ | $G/S \simeq$ $C_2^4$ | |
| Maximal subgroups: | $M_{2,1} \simeq$ $C_2^2.D_4^2$ | $G/M_{2,1} \simeq$ $C_2$ | 16 normal subgroups |
| $M_{2,2} \simeq$ $C_2^5:D_4$ | $G/M_{2,2} \simeq$ $C_2$ | 10 normal subgroups | |
| $M_{2,3} \simeq$ $C_2^5.D_4$ | $G/M_{2,3} \simeq$ $C_2$ | 5 normal subgroups | |
| Maximal quotients: | $m_{2,1} \simeq$ $C_2$ | $G/m_{2,1} \simeq$ $C_2^2.D_4^2$ | 16 normal subgroups |
| $m_{2,2} \simeq$ $C_2$ | $G/m_{2,2} \simeq$ $C_2^3.C_2^5$ | 5 normal subgroups | |
| $m_{2,3} \simeq$ $C_2$ | $G/m_{2,3} \simeq$ $C_2^2.D_4^2$ | 10 normal subgroups |