Finite Group Databasebeta

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Cayley diagram

C2×#1389 #12571

order256
nilpotency class4
isoclinismisoclinic class of #360
automorphismAut(G) order 8192
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C24

Aliases

direct productC2×#1389

Subgroups

subgroupG/H
centerC2×C4#102
commutatorC8C2×C42
Frattini, cosocleC22×C8C23
socleC22#340
prime radicalC24SD16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

285 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);(3,4,7,12,9,13,8,24)(5,14,16,31,19,27,18,33)(6,20,21,29,23,34,22,11)(10,25,26,17,28,32,30,15);(3,5,6,17,7,16,21,32,9,19,23,15,8,18,22,25)(4,10,11,27,24,30,34,31,13,28,29,14,12,26,20,33)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#1389, #146343, #146345, …
maximal normal subgroup ofC4×#1389, #146343, #146345, …
maximal quotient ofC2×#7159, #126666, #126697, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5559);