Finite Group Databasebeta

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Cayley diagram

C2×#13519 #499444

order512
nilpotency class4
isoclinismisoclinic class of #13309
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24

Aliases

direct productC2×#13519

Subgroups

subgroupG/H
centerC2×C4#469
commutator, Frattini, cosocleC22×C8C24
socleC22#2087
prime radicalC24#135
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#499634, C2×#499506, C2×#505707, …
maximal normal subgroup ofC2×#499634, C2×#499506, C2×#505707, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406640);