Finite Group Databasebeta

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Cayley diagram

C22×#1672 #6353582

order512
nilpotency class3
isoclinismisoclinic class of #1665
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC22×#1672

Subgroups

subgroupG/H
center, socle, prime radicalC24#121
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2);(3,4);(6,9)(7,11)(10,12)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22);(5,6,7,9,8,10,11,12);(13,15,17,23)(14,18,20,27)(16,25,26,22)(19,28,24,21)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#20017
maximal normal subgroup ofC22×#20017

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260778);