Finite Group Databasebeta

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Cayley diagram

C2×#19311 #6353967

order512
nilpotency class3
isoclinismisoclinic class of #19304
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#19311

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6363041, C2×#6363072, C2×#6354109
maximal normal subgroup ofC2×#6363041, C2×#6363072, C2×#6354109

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261163);