Finite Group Databasebeta

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Cayley diagram

C2×#1388 #12570

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

Aliases

direct productC2×#1388

Subgroups

subgroupG/H
centerC2×C4#102
commutatorC8C2×C42
Frattini, cosocleC22×C8C23
socleC22#340
prime radicalC24Q16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

277 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);(3,4,7,13,10,15,9,26)(5,16,18,28,21,30,20,34)(6,22,24,31,25,14,8,12)(11,19,27,23,29,33,32,17);(3,5,8,19,9,20,25,17,10,21,24,33,7,18,6,23)(4,11,14,28,13,27,12,30,15,29,22,34,26,32,31,16)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#1388, #146343, #146345, …
maximal normal subgroup ofC4×#1388, #146343, #146345, …
maximal quotient ofC2×#7158, #126664, #126695, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5558);