Finite Group Databasebeta

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Cayley diagram

C4×#1861 #354448

order512
nilpotency class3
isoclinismisoclinic class of #453
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC4×#1861

Subgroups

subgroupG/H
centerC42#120
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C42 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1566 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);(5,6,10,14,11,15,9,17)(7,16,13,12,19,20,18,8);(5,7,10,13,11,19,9,18)(6,12,14,20,15,8,17,16);(5,8)(6,13)(7,17)(9,20)(10,16)(11,12)(14,19)(15,18)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214887, C4×#24472
maximal normal subgroup of#72214887, C4×#24472

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261644);