Finite Group Databasebeta

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Cayley diagram

C22×#421 #60091

order256
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) order 393216
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC22×#421

Subgroups

subgroupG/H
centerC22×C4C2×D4
commutatorC4C24×C4
Frattini, cosocleC2×C4C25
socleC23C2×C22⋊C4
prime radicalC23×#41C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

2599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×#421, S4×#421
maximal subgroup of#123543, #123544, #123546, …
maximal normal subgroup of#123543, #123544, #123546, …
maximal quotient ofC22×#1478, C22×#1479, C22×#1477, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53079);