Finite Group Databasebeta

← Prev (#11918) Next (#11920) →

Cayley diagram

C2×#1358 #11919

order256
nilpotency class4
isoclinismisoclinic class of #1357
automorphismAut(G) order 16384
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#1358

Subgroups

subgroupG/H
center, socleC22#331
commutatorC42C22×C4
Frattini, cosocleC4×C8C23
prime radicalC23#100
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

195 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);(3,4,6,11,7,10,19,26,8,13,20,29,22,12,30,28)(5,14,15,31,16,25,32,34,17,24,33,27,21,9,18,23);(3,5,8,17)(4,9,13,25)(6,18,20,32)(7,21,22,16)(10,24,12,14)(11,27,29,31)(15,19,33,30)(23,26,34,28)

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#94777, C4×#1358, #128917, …
maximal normal subgroup of#94777, C4×#1358, #128917, …
maximal quotient ofC2×#7093, #122858, #122860, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 4907);