Finite Group Databasebeta

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Cayley diagram

C2×#31282 #7504287

order512
nilpotency class3
isoclinismisoclinic class of #31282
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#31282

Subgroups

subgroupG/H
center, socleC22#2684
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#453C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2476 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);(4,9)(5,13)(6,7)(10,14)(11,12)(15,16);(3,4,7,11,8,12,6,9)(5,14,13,18,15,10,16,17);(3,5)(4,10)(6,13)(7,16)(8,15)(9,17)(11,18)(12,14);(4,11)(6,7)(9,12)(10,18)(13,16)(14,17)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7504292, #72250784, #72250788, …
maximal normal subgroup ofC2×#7504292, #72250784, #72250788, …
maximal quotient of#72279625, #72284694, #72284893

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411483);