Finite Group Databasebeta

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Cayley diagram

C2×#13332 #498857

order512
nilpotency class4
isoclinismisoclinic class of #13331
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#13332

Subgroups

subgroupG/H
center, socleC23#459
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#509C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

1620 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)(9,19)(10,15)(11,17)(12,28)(13,18)(14,21)(16,25)(20,29)(22,34)(23,36)(24,35)(26,32)(27,33)(30,38)(31,37);(4,6)(7,9,13,23)(8,14,18,31)(10,24,27,38)(11,20,12,22)(15,19,33,36)(16,34,32,29)(17,35,28,30)(21,26,37,25);(8,15)(9,20)(10,25)(11,12)(14,29)(16,17)(18,33)(19,35)(21,37)(22,23)(26,27)(28,32)(30,36)(31,34)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72281717, #72281756, C2×#499265
maximal normal subgroup of#72281717, #72281756, C2×#499265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406053);