Finite Group Databasebeta

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Cayley diagram

C2×#13414 #499076

order512
nilpotency class4
isoclinismisoclinic class of #13331
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#13414

Subgroups

subgroupG/H
center, socleC23#459
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1258 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)(7,8,13,18)(9,19,22,37)(10,14,12,16)(11,24,27,34)(15,30,33,20)(17,23,35,25)(21,31,28,36)(26,32,38,29);(8,14)(9,20)(10,12)(11,25)(15,31)(16,18)(17,35)(19,29)(21,26)(22,30)(23,27)(28,38)(32,37)(33,36);(4,6)(7,9,10,21,13,22,12,28)(8,15,16,32,18,33,14,29)(11,26,23,20,27,38,25,30)(17,31,34,19,35,36,24,37)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72281975, #72282014, C2×#499265
maximal normal subgroup of#72281975, #72282014, C2×#499265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406272);