Finite Group Databasebeta

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Cayley diagram

C4×#2171 #493214

order512
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4

Aliases

direct productC4×#2171

Subgroups

subgroupG/H
centerC4×C8C2×D4
commutatorC4C22×C4×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radicalC2×#135C2×C4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#12068, #72223664, C8×#2171, …
maximal normal subgroup ofC4×#12068, #72223664, C8×#2171, …
maximal quotient ofC8×#2171, C2×C4×#2171, AGL(1,𝔽5)×#2171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400410);