Finite Group Databasebeta

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Cayley diagram

C4×#1903 #354250

order512
nilpotency class3
isoclinismisoclinic class of #1902
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC4×#1903

Subgroups

subgroupG/H
centerC2×C4#386
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1897 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);(5,6,11,14)(7,10,13,9)(8,15)(12,16)(17,19)(18,20);(5,7,9,6,11,13,10,14)(8,15,17,12,18,20,16,19);(5,8)(6,12)(7,15)(9,17)(10,16)(11,18)(13,20)(14,19)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224749, #72224755, #72224756, …
maximal normal subgroup of#72224749, #72224755, #72224756, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261446);