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Cayley diagram

C22×#2994 #10574348

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC22×#2994

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC22C25×C4
Frattini, cosocleC2×C4C26
socleC24C23×C4
prime radicalC22×#121C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

7210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#18093, C22×#18020, C22×#60305
maximal normal subgroup ofC22×#18093, C22×#18020, C22×#60305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481544);