Finite Group Databasebeta

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Cayley diagram

C2×#13181 #495062

order512
nilpotency class4
isoclinismisoclinic class of #2212
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#13181

Subgroups

subgroupG/H
center, socleC23#459
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(3,4,6,5)(8,14)(9,10)(11,12)(15,22)(16,21)(17,18)(19,20);(3,5,6,4)(7,8)(9,14)(10,19)(11,15)(12,20)(13,16)(17,22)(18,21);(7,9,12,17,13,18,11,10)(8,14,20,22,16,21,15,19)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×#2213, #72242485
maximal normal subgroup ofC2×C4×#2213, #72242485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402258);