Finite Group Databasebeta

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Cayley diagram

C2×#10685 #243951

order512
nilpotency class3
isoclinismisoclinic class of #10304
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#10685

Subgroups

subgroupG/H
center, socle, prime radicalC24#127
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#244406, #72214416
maximal normal subgroup ofC2×#244406, #72214416
maximal quotient ofC2×#121387

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151147);