Finite Group Databasebeta

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Cayley diagram

C2×#1423 #13631

order256
nilpotency class5
isoclinismisoclinic class of D32
automorphismAut(G) order 32768
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#1423

Subgroups

subgroupG/H
center, socle, prime radicalC23D16
commutatorC16C22×C4
Frattini, cosocleC2×C16C23
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2);(3,4,5,6)(8,13)(9,14)(10,18)(11,21)(15,30)(16,33)(17,26)(19,35)(20,29)(22,34)(23,24)(25,31)(27,28)(32,38)(36,37);(7,8,14,24,10,15,27,33,21,31,35,38,34,36,20,26,12,17,29,37,22,32,19,25,11,16,28,30,18,23,9,13)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#94761, #94768, C4×#1423, …
maximal normal subgroup of#94761, #94768, C4×#1423, …
maximal quotient ofC2×#7460, #151050, #151061, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6619);