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Cayley diagram

C2×C64 #1427

order128
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×#504 (order 128) · Inn(G) = C1 · Out(G) = C2×#504
homologyAbelianization (H1) = G · Schur multiplier (H2) = C2

Aliases

abelianC2×C64
multiplicative(ℤ/256)ˣ

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC32C22
socle, prime radicalC22C32
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 66
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 2 generators on 66 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(65,66)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

center of#94814, C22⋊C128, C4⋊C128, …
Frattini subgroup ofC4×C128, #94814, #94815, …
maximal subgroup ofC4×C64, #7510, C22⋊C64, …
maximal normal subgroup ofC4×C64, #7510, C22⋊C64, …
maximal quotient ofC4×C64, C22⋊C64, C4⋊C64, …
automorphism group ofC256
outer automorphism group ofC256

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 159);