Finite Group Databasebeta

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Cayley diagram

C23×Q82 #10585935

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C27 · Schur multiplier (H2) = C219

Aliases

direct productC23×Q82

Subgroups

subgroupG/H
center, socle, prime radicalC25C24
commutator, Frattini, cosocleC22C27
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C22 ▷ C1

37970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×C22⋊(Q8⋊SL(2,𝔽3))
maximal subgroup ofC23×#2205, C2×#1421181, C2×C22⋊(Q8⋊SL(2,𝔽3))
maximal normal subgroup ofC23×#2205, C2×#1421181, C2×C22⋊(Q8⋊SL(2,𝔽3))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493131);