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Properties

Label 512.7591302
Order \( 2^{9} \)
Exponent \( 2^{2} \)
Nilpotent yes
Solvable yes
$\card{G^{\mathrm{ab}}}$ \( 2^{6} \)
$\card{Z(G)}$ \( 2^{3} \)
$\card{\Aut(G)}$ \( 2^{20} \)
$\card{\mathrm{Out}(G)}$ \( 2^{14} \)
Trans deg. not computed
Rank $6$

Learn more

This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.

Group information

Description:$C_2^3 . C_2^6$
Order: \(512\)\(\medspace = 2^{9} \)
Exponent: \(4\)\(\medspace = 2^{2} \)
Automorphism group:Group of order 1048576
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 1 199 312 512
Conjugacy classes   1 56 56 113
Divisions data not computed
Autjugacy classes data not computed

Dimension 1 2 4 8
Irr. complex chars.   64 32 16 1 113

Constructions

Presentation: ${\langle a, b, c, d, e, f, g, h, i \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$ Copy content Toggle raw display

Homology

Abelianization: $C_{2}^{6} $

Subgroups

Center: $Z \simeq$ $C_2^3$ $G/Z \simeq$ $C_2^6$
Commutator: $G' \simeq$ $C_2^3$ $G/G' \simeq$ $C_2^6$
Frattini: $\Phi \simeq$ $C_2^3$ $G/\Phi \simeq$ $C_2^6$
Fitting: $\operatorname{Fit} \simeq$ $C_2^3 . C_2^6$ $G/\operatorname{Fit} \simeq$ $C_1$
Radical: $R \simeq$ $C_2^3 . C_2^6$ $G/R \simeq$ $C_1$
Socle: $S \simeq$ $C_2^3$ $G/S \simeq$ $C_2^6$
Maximal subgroups: $M_{2,1} \simeq$ $D_4^2:C_4$ $G/M_{2,1} \simeq$ $C_2$ 2 normal subgroups
$M_{2,2} \simeq$ $C_2^5:D_4$ $G/M_{2,2} \simeq$ $C_2$ 2 normal subgroups
$M_{2,3} \simeq$ $C_4.D_4^2$ $G/M_{2,3} \simeq$ $C_2$
$M_{2,4} \simeq$ $C_2^3.C_2^5$ $G/M_{2,4} \simeq$ $C_2$
$M_{2,5} \simeq$ $C_2^3.C_2^5$ $G/M_{2,5} \simeq$ $C_2$ 2 normal subgroups
$M_{2,6} \simeq$ $C_2^3.C_2^5$ $G/M_{2,6} \simeq$ $C_2$
$M_{2,7} \simeq$ $C_2^2:D_4^2$ $G/M_{2,7} \simeq$ $C_2$ 6 normal subgroups
$M_{2,8} \simeq$ $C_4.D_4^2$ $G/M_{2,8} \simeq$ $C_2$ 2 normal subgroups
$M_{2,9} \simeq$ $C_2^3.C_2^5$ $G/M_{2,9} \simeq$ $C_2$
$M_{2,10} \simeq$ $C_2^3.C_2^5$ $G/M_{2,10} \simeq$ $C_2$ 2 normal subgroups
$M_{2,11} \simeq$ $C_2^5:C_2^3$ $G/M_{2,11} \simeq$ $C_2$ 4 normal subgroups
$M_{2,12} \simeq$ $C_4.D_4^2$ $G/M_{2,12} \simeq$ $C_2$ 2 normal subgroups
$M_{2,13} \simeq$ $C_2^2.D_4^2$ $G/M_{2,13} \simeq$ $C_2$ 4 normal subgroups
$M_{2,14} \simeq$ $C_4.D_4^2$ $G/M_{2,14} \simeq$ $C_2$ 2 normal subgroups
$M_{2,15} \simeq$ $C_2^2.D_4^2$ $G/M_{2,15} \simeq$ $C_2$ 2 normal subgroups
$M_{2,16} \simeq$ $C_2^2.D_4^2$ $G/M_{2,16} \simeq$ $C_2$ 4 normal subgroups
$M_{2,17} \simeq$ $C_2^2.D_4^2$ $G/M_{2,17} \simeq$ $C_2$ 2 normal subgroups
$M_{2,18} \simeq$ $C_4.D_4^2$ $G/M_{2,18} \simeq$ $C_2$
$M_{2,19} \simeq$ $C_4.D_4^2$ $G/M_{2,19} \simeq$ $C_2$ 2 normal subgroups
$M_{2,20} \simeq$ $C_4.D_4^2$ $G/M_{2,20} \simeq$ $C_2$
$M_{2,21} \simeq$ $C_2^2:D_4^2$ $G/M_{2,21} \simeq$ $C_2$ 4 normal subgroups
$M_{2,22} \simeq$ $C_2^2.D_4^2$ $G/M_{2,22} \simeq$ $C_2$ 2 normal subgroups
$M_{2,23} \simeq$ $C_4:D_4^2$ $G/M_{2,23} \simeq$ $C_2$
$M_{2,24} \simeq$ $C_4.D_4^2$ $G/M_{2,24} \simeq$ $C_2$ 2 normal subgroups
$M_{2,25} \simeq$ $C_2^2.D_4^2$ $G/M_{2,25} \simeq$ $C_2$ 4 normal subgroups
$M_{2,26} \simeq$ $C_4:D_4^2$ $G/M_{2,26} \simeq$ $C_2$ 2 normal subgroups
$M_{2,27} \simeq$ $C_2^2\times D_4^2$ $G/M_{2,27} \simeq$ $C_2$ 2 normal subgroups
$M_{2,28} \simeq$ $C_4^2:C_2^4$ $G/M_{2,28} \simeq$ $C_2$ 2 normal subgroups
Maximal quotients: $m_{2,1} \simeq$ $C_2$ $G/m_{2,1} \simeq$ $C_2^2\times D_4^2$
$m_{2,2} \simeq$ $C_2$ $G/m_{2,2} \simeq$ $C_4^2:C_2^4$ 2 normal subgroups
$m_{2,3} \simeq$ $C_2$ $G/m_{2,3} \simeq$ $D_4^2:C_2^2$
$m_{2,4} \simeq$ $C_2$ $G/m_{2,4} \simeq$ $D_4^2:C_2^2$ 2 normal subgroups
$m_{2,5} \simeq$ $C_2$ $G/m_{2,5} \simeq$ $D_4^2:C_2^2$