Finite Group Databasebeta

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Cayley diagram

C2×#11483 #480122

order512
nilpotency class4
isoclinismisoclinic class of AΣL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C26

Aliases

direct productC2×#11483

Subgroups

subgroupG/H
centerC2×C8#99
commutatorC23C23×C8
Frattini, cosocleC2×M16C24
socleC22#1459
prime radicalC2×#525C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C22×#131 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

1048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#480440, #72239959, #72239963, …
maximal normal subgroup ofC2×#480440, #72239959, #72239963, …
maximal quotient of#72239994, #72240701, #72308032

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387318);