Finite Group Databasebeta

← Prev (#20412) Next (#20414) →

Cayley diagram

C4×#419 #20413

order256
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) order 131072
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC4×#419

Subgroups

subgroupG/H
centerC22×C4C2×D4
commutatorC4C22×C42
Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC24C2×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,3,4);(5,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(9,11,14,21)(10,15,17,23)(12,20,13,19)(16,24,22,18);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#105649, #105651, #106419, …
maximal normal subgroup of#105649, #105651, #106419, …
maximal quotient ofC4×#1479, #182158, #182171, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13401);