Finite Group Databasebeta

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Cayley diagram

C22×D4×M16 #10574192

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C214

Aliases

direct productC22×D4×M16

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC22C25×C4
Frattini, cosocleC2×C4C26
socleC24C23×C4
prime radicalC24×D4C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

12682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 16 points:

(1,2);(3,4);(5,6,7,8);(6,8);(9,10,11,12,13,14,15,16);(10,14)(12,16)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×#407, C22×D4×AGL(1,ℤ/8), C2×D4×#418, …
maximal normal subgroup ofC2×D4×#407, C22×D4×AGL(1,ℤ/8), C2×D4×#418, …
maximal quotient ofC2×D4×#407, C2×#6373373, C2×#6376471, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481388);