Finite Group Databasebeta

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Cayley diagram

C2×#1422 #13629

order256
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×#72285688 (order 2048) · Inn(G) = D4 · Out(G) = C2×C4×#120
homologyAbelianization (H1) = C22×C16 · Schur multiplier (H2) = C22

Aliases

direct productC2×#1422

Subgroups

subgroupG/H
centerC2×C16D4
commutatorC4C22×C16
Frattini, cosocleC2×C16C23
socleC22C4⋊C16
prime radicalC23M32
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C16 ◁ C22×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

107 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);(3,4,7,13,8,14,22,29,9,12,23,27,19,28,32,34,10,15,24,30,17,11,25,26,6,20,21,33,5,16,18,31);(3,5,6,17,10,19,9,8)(4,11,12,16,15,14,20,28)(7,18,21,25,24,32,23,22)(13,26,27,31,30,29,33,34)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#94018, #94019, #94020, …
maximal normal subgroup of#94018, #94019, #94020, …
maximal quotient ofC2×#7455, #151019, C2×#7458, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6617);