Finite Group Databasebeta

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Cayley diagram

C2×#13458 #499206

order512
nilpotency class4
isoclinismisoclinic class of #13371
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#13458

Subgroups

subgroupG/H
center, socleC23#476
commutator, Frattini, cosocleC22×C8C24
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

826 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);(7,8,10,14)(9,19)(11,25,24,35)(12,18,13,17)(15,31)(16,27,30,28)(20,34)(21,29)(22,36)(23,37)(26,32)(33,38);(3,4)(5,6)(8,14)(9,20,23,22)(15,32,29,33)(16,30)(17,18)(19,34,37,36)(21,38,31,26)(25,35);(4,6)(7,9,12,22,10,23,13,20)(8,15,17,32,14,29,18,33)(11,26,27,31,24,38,28,21)(16,34,35,37,30,36,25,19)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72281594, #72281818, C2×#499265
maximal normal subgroup of#72281594, #72281818, C2×#499265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406402);