Finite Group Databasebeta

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Cayley diagram

C2×#18141 #6346382

order512
nilpotency class3
isoclinismisoclinic class of #18132
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#18141

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

3260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268065, #72268066, C2×#11715588
maximal normal subgroup of#72268065, #72268066

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253578);