Finite Group Databasebeta

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Cayley diagram

C2×C4×D42 #7625387

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C215

Aliases

direct productC2×C4×D42

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC22C25×C4
Frattini, cosocleC23C26
socleC24C25
prime radicalC22×D42C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23 ▷ C1

19286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 14 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D42×AGL(1,𝔽5), C2×#209860, C2×D4×#442, …
maximal normal subgroup ofC2×#209860, C2×D4×#442, #72231276, …
maximal quotient of#72214822, C4×#61110, C2×D42×AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532583);