Finite Group Databasebeta

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Cayley diagram

C2×#7180 #126790

order512
nilpotency class4
isoclinismisoclinic class of #365
automorphism?
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24

Aliases

direct productC2×#7180

Subgroups

subgroupG/H
centerC2×C8#106
commutatorC8C2×C4×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1279
prime radicalC24#104
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226839, #72226841, C2×#137561, …
maximal normal subgroup of#72226839, #72226841, C2×#137561, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33986);