Finite Group Databasebeta

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Cayley diagram

C4×#2008 #359069

order512
nilpotency class3
isoclinismisoclinic class of #2008
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC4×#2008

Subgroups

subgroupG/H
centerC2×C4#392
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#392
prime radicalC2×#2008C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2071 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)(7,15)(8,13)(9,12)(10,14)(11,16)(17,20)(18,19);(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19);(6,12)(8,9)(11,15)(13,14)(16,18)(19,20)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#2008, C4×#23882, C4×#23872, …
maximal normal subgroup ofC4×#23882, C4×#23872, #72224743, …
maximal quotient ofAGL(1,𝔽5)×#2008, #72257884, #72264012, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266265);