Finite Group Databasebeta

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Cayley diagram

C2×#13733 #513059

order512
nilpotency class5
isoclinismisoclinic class of #13694
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C23×C4

Aliases

direct productC2×#13733

Subgroups

subgroupG/H
center, socle, prime radicalC23C2×D16
commutator, Frattini, cosocleC2×C16C24
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4,6,8)(5,9,10,7)(12,17)(13,22)(14,23)(15,26)(18,25)(19,34)(20,37)(21,31)(24,33)(27,40)(28,29)(30,35)(32,41)(36,42)(38,39);(3,5,6,10)(4,7,8,9);(11,12,18,32,14,19,24,30,26,35,39,34,40,41,13,17,16,21,22,29,27,36,38,37,15,20,33,42,23,28,25,31)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#513082
maximal normal subgroup ofC2×#513082

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420255);