Finite Group Databasebeta

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Cayley diagram

C4×#1792 #354823

order512
nilpotency class3
isoclinismisoclinic class of #453
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4

Aliases

direct productC4×#1792

Subgroups

subgroupG/H
centerC22×C4#120
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

4068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239475, C4×#13343, AGL(1,𝔽5)×#1792, …
maximal normal subgroup of#72239475, C4×#13343, C4×#32888, …
maximal quotient ofC4×#8440, AGL(1,𝔽5)×#1792, C4×#8453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262019);