Finite Group Databasebeta

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Cayley diagram

C2×#62818 #10586165

order512
nilpotency class2
isoclinismisoclinic class of #62817
automorphism?
homologyAbelianization (H1) = C27 · Schur multiplier (H2) = C219

Aliases

direct productC2×#62818

Subgroups

subgroupG/H
center, socleC23C26
commutator, Frattini, cosocleC22C27
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C22 ▷ C1

38618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 7 generators on 18 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10586472, #72234454, #72234456, …
maximal normal subgroup ofC2×#10586472, #72234454, #72234456, …
maximal quotient of#72233590, #72233694, C2×#7744685, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493361);