Finite Group Databasebeta

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Cayley diagram

C2×#23994 #6451773

order512
nilpotency class3
isoclinismisoclinic class of #23994
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23994

Subgroups

subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC22×C4C25
socleC22#2404
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72216067, C2×#6499855, #72219644, …
maximal normal subgroup of#72216067, C2×#6499855, #72219644, …
maximal quotient of#72216067, #72233068, #72233848, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358969);