Finite Group Databasebeta

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Cayley diagram

C2×#35956 #7629428

order512
nilpotency class2
isoclinismisoclinic class of #35680
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#35956

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
prime radicalC2×#2657C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

8322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7690378, C2×#7691803, C2×#7629439
maximal normal subgroup ofC2×#7690378, C2×#7691803, C2×#7629439

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536624);