Finite Group Databasebeta

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Cayley diagram

C4×#1526 #182693

order512
nilpotency class3
isoclinismisoclinic class of #327
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC4×#1526

Subgroups

subgroupG/H
centerC2×C42C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C42 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239481, #72239496
maximal normal subgroup of#72239481, #72239496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89889);