Abstract group 512.1894: $(C_2^2\times Q_8).Q_{16}$

• Label: 512.1894
• Order: \( 2^{9} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $32$
• Trans deg.: $32$
• Rank: $2$

Group information

Description:	$(C_2^2\times Q_8).Q_{16}$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2\times C_2^5.C_2.C_2^4$, of order \(2048\)\(\medspace = 2^{11} \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$6$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8
Elements	1	31	288	192	512
Conjugacy classes	1	7	15	6	29
Divisions	1	7	10	2	20
Autjugacy classes	1	6	10	3	20

Dimension	1	2	4	8	16
Irr. complex chars.	8	10	5	6	0	29
Irr. rational chars.	4	4	7	3	2	20

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$32$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	8	16	16

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{4}=b^{4}=c^{2}=d^{2}=e^{8}=[b,d]=[c,d]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $32$ $\langle(1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,29,13,25)(10,30,14,26)(11,31,15,27) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^3.C_2^4)$ $\,\rtimes\,$ $C_4$	$(C_2^3.C_4^2)$ $\,\rtimes\,$ $C_4$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_4:C_2^3)$ . $D_4$	$(C_2^3.D_4)$ . $D_4$	$(C_2^2\times Q_8)$ . $Q_{16}$	$C_2$ . $(C_2^4.Q_{16})$	all 18

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 1298 subgroups in 209 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^4.Q_{16}$
Commutator:	$G' \simeq$ $C_2^3.D_4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^4.C_2^3$	$G/\Phi \simeq$ $C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2^2\times Q_8).Q_{16}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_2^2\times Q_8).Q_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^4.Q_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2^2\times Q_8).Q_{16}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 512: $(C_2^2\times Q_8).Q_{16}$

Order 256: $(C_2\times C_4^2).D_4$, $(C_2^2\times D_4):C_8$, $(C_2^2\times C_4).C_4^2$

Order 128: $C_2^4.C_2^3$ x 2, $C_2^4.D_4$, $C_2^3.\OD_{16}$, $C_2^3.C_4^2$, $C_2^4.D_4$, $C_2^3.C_2^4$

Order 64: $C_2^3.D_4$ x 4, $C_2^4:C_4$ x 3, $C_4^2:C_2^2$ x 2, $C_2^2:\OD_{16}$, $C_2^3.Q_8$, $C_2^3.C_2^3$, $C_2^3.C_2^3$, $C_2^2:C_4^2$, $C_2^3:C_8$, $D_4:C_2^3$

Order 32: $C_2^3:C_4$ x 6, $C_2.C_4^2$ x 6, $C_2^3:C_4$ x 5, $D_4:C_2^2$ x 4, $C_4\wr C_2$ x 4, $C_2^2:C_8$ x 3, $C_2^2.D_4$ x 3, $C_2^2\times D_4$ x 2, $C_2^3\times C_4$ x 2, $C_2\times C_4^2$ x 2, $C_2^2\times Q_8$, $C_2\times \OD_{16}$

Order 16: $C_2^2\times C_4$ x 16, $D_4:C_2$ x 10, $C_2^2:C_4$ x 8, $C_2\times D_4$ x 8, $C_4:C_4$ x 5, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_4^2$ x 2, $C_2^4$ x 2, $C_2\times Q_8$ x 2

Order 8: $C_2\times C_4$ x 32, $C_2^3$ x 10, $D_4$ x 7, $Q_8$ x 3, $C_8$ x 2

Order 4: $C_2^2$ x 14, $C_4$ x 10

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $(C_2^2\times Q_8).Q_{16}$

Order 256: $(C_2\times C_4^2).D_4$, $(C_2^2\times D_4):C_8$, $(C_2^2\times C_4).C_4^2$

Order 128: $C_2^4.C_2^3$ x 2, $C_2^4.D_4$, $C_2^3.\OD_{16}$, $C_2^3.C_4^2$, $C_2^4.D_4$, $C_2^3.C_2^4$

Order 64: $C_2^4:C_4$ x 3, $C_2^3.D_4$ x 3, $C_4^2:C_2^2$ x 2, $C_2^2:\OD_{16}$, $C_2^3.Q_8$, $C_2^3.C_2^3$, $C_2^3.C_2^3$, $C_2^2:C_4^2$, $C_2^3:C_8$, $D_4:C_2^3$

Order 32: $C_2.C_4^2$ x 5, $C_2^3:C_4$ x 4, $C_2^3:C_4$ x 4, $C_2^2:C_8$ x 3, $D_4:C_2^2$ x 3, $C_2^2.D_4$ x 3, $C_2^2\times D_4$ x 2, $C_2^3\times C_4$ x 2, $C_2\times C_4^2$ x 2, $C_4\wr C_2$ x 2, $C_2^2\times Q_8$, $C_2\times \OD_{16}$

Order 16: $C_2^2\times C_4$ x 14, $D_4:C_2$ x 6, $C_2\times D_4$ x 6, $C_2^2:C_4$ x 5, $C_4:C_4$ x 3, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_4^2$ x 2, $C_2^4$ x 2, $C_2\times Q_8$ x 2

Order 8: $C_2\times C_4$ x 26, $C_2^3$ x 9, $D_4$ x 4, $Q_8$ x 2, $C_8$ x 2

Order 4: $C_2^2$ x 12, $C_4$ x 8

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $(C_2^2\times Q_8).Q_{16}$ ($C_1$)

Order 256: $(C_2\times C_4^2).D_4$ ($C_2$), $(C_2^2\times D_4):C_8$ ($C_2$), $(C_2^2\times C_4).C_4^2$ ($C_2$)

Order 128: $C_2^4.C_2^3$ ($C_2^2$), $C_2^3.C_4^2$ ($C_4$), $C_2^3.C_2^4$ ($C_4$)

Order 64: $C_2^3.D_4$ ($D_4$), $C_2^3.D_4$ ($C_2\times C_4$), $D_4:C_2^3$ ($D_4$)

Order 32: $C_2^2\times Q_8$ ($Q_{16}$), $C_2^2\times D_4$ ($\SD_{16}$), $C_2^3\times C_4$ ($C_2^2:C_4$)

Order 16: $C_2^4$ ($C_2^3:C_4$), $C_2^2\times C_4$ ($C_4\wr C_2$), $C_2^2\times C_4$ ($Q_8:C_4$)

Order 8: $C_2^3$ ($C_2^2.Q_{16}$), $C_2^3$ ($C_4^2:C_4$), $C_2^3$ ($C_2\wr C_4$)

Order 4: $C_2^2$ ($C_2^3.Q_{16}$)

Order 2: $C_2$ ($C_2^4.Q_{16}$)

Order 1: $C_1$ ($(C_2^2\times Q_8).Q_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $(C_2^2\times Q_8).Q_{16}$ ($C_1$)

Order 256: $(C_2\times C_4^2).D_4$ ($C_2$), $(C_2^2\times D_4):C_8$ ($C_2$), $(C_2^2\times C_4).C_4^2$ ($C_2$)

Order 128: $C_2^4.C_2^3$ ($C_2^2$), $C_2^3.C_4^2$ ($C_4$), $C_2^3.C_2^4$ ($C_4$)

Order 64: $C_2^3.D_4$ ($D_4$), $C_2^3.D_4$ ($C_2\times C_4$), $D_4:C_2^3$ ($D_4$)

Order 32: $C_2^2\times Q_8$ ($Q_{16}$), $C_2^2\times D_4$ ($\SD_{16}$), $C_2^3\times C_4$ ($C_2^2:C_4$)

Order 16: $C_2^4$ ($C_2^3:C_4$), $C_2^2\times C_4$ ($C_4\wr C_2$), $C_2^2\times C_4$ ($Q_8:C_4$)

Order 8: $C_2^3$ ($C_2^2.Q_{16}$), $C_2^3$ ($C_4^2:C_4$), $C_2^3$ ($C_2\wr C_4$)

Order 4: $C_2^2$ ($C_2^3.Q_{16}$)

Order 2: $C_2$ ($C_2^4.Q_{16}$)

Order 1: $C_1$ ($(C_2^2\times Q_8).Q_{16}$)

Series

Derived series

$(C_2^2\times Q_8).Q_{16}$

$\rhd$

$C_2^3.D_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$(C_2^2\times Q_8).Q_{16}$

$\rhd$

$(C_2\times C_4^2).D_4$

$\rhd$

$C_2^4.C_2^3$

$\rhd$

$C_2^3.D_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times Q_8).Q_{16}$

$\rhd$

$C_2^3.D_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^4$

$\lhd$

$C_2^3\times C_4$

$\lhd$

$C_2^4.C_2^3$

$\lhd$

$(C_2^2\times Q_8).Q_{16}$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $29 \times 29$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.