Finite Group Databasebeta

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Cayley diagram

C2×#1369 #12001

order256
nilpotency class4
isoclinismisoclinic class of #365
automorphismAut(G) order 4096
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C22

Aliases

direct productC2×#1369

Subgroups

subgroupG/H
centerC2×C4#106
commutatorC8C22×C8
Frattini, cosocleC4×C8C23
socleC22#334
prime radicalC23#108
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C4×C8 ▷ C2×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);(3,4,7,13,9,14,24,30,10,15,23,12,8,26,6,22)(5,16,18,27,20,31,17,28,21,29,25,34,19,33,32,11);(3,5,6,17,8,19,23,18,10,21,24,32,9,20,7,25)(4,11,12,16,14,27,22,31,15,28,13,29,26,34,30,33)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#94030, #94031, #94058, …
maximal normal subgroup of#94030, #94031, #94058, …
maximal quotient ofC2×#7099, #122919, #122921, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4989);