Finite Group Databasebeta

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Cayley diagram

C22×#1484 #6342941

order512
nilpotency class3
isoclinismisoclinic class of #447
automorphismAut(G) order 12582912
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC22×#1484

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC24C2×C22⋊C4
prime radicalC22×#532C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

4678 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);(5,6)(7,11)(8,10)(9,12)(13,14,17,20,18,15,16,19);(6,10)(7,8)(11,12)(13,15)(14,18)(16,20)(17,19);(5,7,9,8)(6,11,12,10)(13,16,18,17)(14,19,15,20)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#18091, C22×#17917, C22×#17517
maximal normal subgroup ofC22×#18091, C22×#17917, C22×#17517

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250137);