Finite Group Databasebeta

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Cayley diagram

C2×#1424 #13632

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

Aliases

direct productC2×#1424

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#94764, #94767, C4×#1424, …
maximal normal subgroup of#94764, #94767, C4×#1424, …
maximal quotient ofC2×#7461, #151051, #151062, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6620);