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Cayley diagram

Q162 #20009

order256
nilpotency class3
isoclinismisoclinic class of D82
automorphismAut(G) order 32768
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24

Aliases

direct productQ162

Subgroups

subgroupG/H
center, socle, prime radicalC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16);(17,18,19,20,21,22,23,24)(25,32,31,30,29,28,27,26);(17,25,21,29)(18,26,22,30)(19,27,23,31)(20,28,24,32)

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#123279, #123289, #123301, …
maximal normal subgroup of#123279, #123289, #123301, …
maximal quotient ofQ16×#103, Q16×#107, #227021, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12997);