Finite Group Databasebeta

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Cayley diagram

C2×#18137 #6346378

order512
nilpotency class3
isoclinismisoclinic class of #18132
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#18137

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#534C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6347092
maximal normal subgroup ofC2×#6347092

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253574);