Finite Group Databasebeta

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Cayley diagram

C4×#1910 #354224

order512
nilpotency class3
isoclinismisoclinic class of #1909
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42

Aliases

direct productC4×#1910

Subgroups

subgroupG/H
centerC2×C4#386
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC2×#535C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#1910, C4×#23882, #72224746, …
maximal normal subgroup ofC4×#23882, #72224746, #72224755, …
maximal quotient ofAGL(1,𝔽5)×#1910, #72239304, #72264015, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261420);