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Cayley diagram

C2×#1356 #11910

order256
nilpotency class4
isoclinismisoclinic class of #1355
automorphismAut(G) = #72306524 (order 4096) · Inn(G) = #329 · Out(G) = C2×#120
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×#1356

Subgroups

subgroupG/H
center, socleC22#329
commutatorC42C22×C4
Frattini, cosocle#97C23
prime radicalC23#100
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G
LECSG ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1

157 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);(3,4,6,11,7,12,20,30,8,13,19,31,22,10,27,28)(5,14,15,32,16,9,23,24,17,25,33,29,21,26,18,34);(3,5,8,17)(4,9,13,26)(6,18,19,23)(7,21,22,16)(10,25,12,14)(11,29,31,32)(15,20,33,27)(24,28,34,30)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#94785, C4×#1356, #128889, …
maximal normal subgroup of#94785, C4×#1356, #128889, …
maximal quotient ofC2×#7117, C2×#7130, #124206, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4898);