Finite Group Databasebeta

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Cayley diagram

C2×#24117 #6448565

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#24117

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7520904, C2×#7519680, C2×#6512638
maximal normal subgroup ofC2×#7520904, C2×#7519680, C2×#6512638

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355761);