Finite Group Databasebeta

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Cayley diagram

C2×C4×#347 #492829

order512
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphism?
homologyAbelianization (H1) = C2×C43 · Schur multiplier (H2) = C23×C42

Aliases

direct productC2×C4×#347

Subgroups

subgroupG/H
centerC2×C42C42
commutatorC4C2×C43
Frattini, cosocleC22×C8C24
socleC23C2×#97
prime radicalC24C2×M16
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C2×C42 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×AGL(1,ℤ/16), C2×#493177, C4×#12647, …
maximal normal subgroup ofC2×C4×AGL(1,ℤ/16), C2×#493177, C4×#12647, …
maximal quotient ofC2×C4×#1313, C2×C8×#347, C2×#126544, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400025);