Finite Group Databasebeta

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Cayley diagram

C2×D4×#126 #7629730

order512
nilpotency class2
isoclinismisoclinic class of D4×#120
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×D4×#126

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

9126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 22 points:

(1,2);(3,4,5,6);(4,6);(8,12)(9,16)(14,21)(15,19)(17,20)(18,22);(7,8,13,19)(9,14,20,21)(10,12,11,15)(16,22,17,18);(7,9,10,17)(8,14,12,18)(11,16,13,20)(15,22,19,21)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×#529
maximal normal subgroup ofC2×D4×#529

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536926);