Finite Group Databasebeta

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Cayley diagram

C2×#10659 #243824

order512
nilpotency class3
isoclinismisoclinic class of #10012
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#10659

Subgroups

subgroupG/H
centerC22×C4#127
commutator, Frattini, cosocleC2×C42C24
socleC23#399
prime radicalC23×D4Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#244127, C2×#244113, C2×#243831
maximal normal subgroup ofC2×#244127, C2×#244113, C2×#243831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151020);