Finite Group Databasebeta

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Cayley diagram

C22×#1628 #6348035

order512
nilpotency class3
isoclinismisoclinic class of #1619
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC22×#1628

Subgroups

subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC42C25
prime radicalC22×#459C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

3474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#19855, C22×#19871
maximal normal subgroup ofC22×#19855, C22×#19871

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255231);