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Cayley diagram

C22×#1600 #6346351

order512
nilpotency class3
isoclinismisoclinic class of #1595
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC22×#1600

Subgroups

subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

6158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#18162
maximal normal subgroup ofC22×#18162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253547);