Finite Group Databasebeta

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Cayley diagram

C2×#13317 #498819

order512
nilpotency class4
isoclinismisoclinic class of #13316
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#13317

Subgroups

subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#3054C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282210, #72282251, C2×#499265
maximal normal subgroup of#72282210, #72282251, C2×#499265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406015);