Finite Group Databasebeta

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Cayley diagram

C2×#24082 #6452945

order512
nilpotency class3
isoclinismisoclinic class of #23876
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24082

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3494C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6453878, C2×#6453869, C2×#6452952
maximal normal subgroup ofC2×#6453878, C2×#6453869, C2×#6452952
maximal quotient ofC2×#257014, #72309232

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6360141);