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Cayley diagram

C2×#350 #2116

order128
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×#62773 (order 512) · Inn(G) = D4 · Out(G) = C22×#41
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC2×#350

Subgroups

subgroupG/H
centerC2×C8D4
commutatorC4C22×C8
Frattini, cosocleC2×C8C23
socleC22C22⋊C8
prime radicalC2×#41C4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

112 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);(3,4,7,12,8,13,22,30,9,14,23,31,6,11,20,28)(5,15,17,32,16,26,33,29,18,27,21,34,19,10,24,25);(3,5)(4,10)(6,19)(7,21)(8,16)(9,18)(11,27)(12,29)(13,15)(14,26)(17,23)(20,33)(22,24)(25,31)(28,32)(30,34)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#350, #12604, #12614, …
maximal normal subgroup ofC4×#350, #12604, #12614, …
maximal quotient ofC2×#1315, #11422, C2×#1323, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 848);