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Cayley diagram

C2×#1398 #13550

order256
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphismAut(G) = #72286249 (order 2048) · Inn(G) = C42 · Out(G) = C22×#120
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22

Aliases

direct productC2×#1398

Subgroups

subgroupG/H
centerC2×C8C42
commutatorC4C2×C4×C8
Frattini, cosocleC2×C16C23
socleC22#346
prime radicalC23M32
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C2×C8 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

111 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);(3,4,5,10,6,12,15,25,7,13,16,23,17,27,29,33,8,11,14,24,18,26,28,34,19,9,20,21,30,22,31,32);(4,9,11,13)(5,14)(10,23,24,21)(12,22,26,27)(15,28)(16,20)(25,33,34,32)(29,31)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#93995, #93996, #93997, …
maximal normal subgroup of#93995, #93996, #93997, …
maximal quotient ofC2×#7330, #149495, C2×#7332, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6538);