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Cayley diagram

C2×M64 #2257

order128
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#21483 (order 512) · Inn(G) = C22 · Out(G) = C22×C4×D4
homologyAbelianization (H1) = C22×C16 · Schur multiplier (H2) = C22

Aliases

direct productC2×M64

Subgroups

subgroupG/H
centerC2×C16C22
commutatorC2C22×C16
Frattini, cosocleC16C23
socleC22C2×C16
prime radicalC23C16
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C16 ◁ G
LECSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

56 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);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34);(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(18,34)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#94512, #94513, #94577, …
maximal subgroup of#7466, #7467, #7468, …
maximal normal subgroup of#7466, #7467, #7468, …
maximal quotient ofC2×#1397, C4×M64, C2×C22⋊C32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 989);