Finite Group Databasebeta

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Cayley diagram

C2×#1919 #21030

order256
nilpotency class3
isoclinismisoclinic class of #447
automorphismAut(G) order 131072
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#1919

Subgroups

subgroupG/H
center, socle, prime radicalC24C2×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(3,4,6,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22);(3,5,6,4)(8,11)(12,14)(16,19)(20,21);(7,9,10,13)(8,12,11,14)(16,20)(18,22)(19,21)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#99893, #99909, C4×#1919, …
maximal normal subgroup of#99893, #99909, C4×#1919, …
maximal quotient ofC2×#9534, C2×#9538, #202206, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14018);