Finite Group Databasebeta

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Cayley diagram

C2×#44877 #7669446

order512
nilpotency class2
isoclinismisoclinic class of #42996
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C214

Aliases

direct productC2×#44877

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

15382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7684106, #72232137, #72232152, …
maximal normal subgroup ofC2×#7684106, #72232137, #72232152, …
maximal quotient of#72232084, #72232133, C2×#1418391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7576642);