Finite Group Databasebeta

← Prev (#6380075) Next (#6380077) →

Cayley diagram

D4×#428 #6380076

order512
nilpotency class3
isoclinismisoclinic class of D4×AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productD4×#428

Subgroups

subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 16 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D4×AGL(1,ℤ/8), #72279582, #72279630, …
maximal normal subgroup ofC4×D4×AGL(1,ℤ/8), #72279582, #72279630, …
maximal quotient of#72279630, #72280022, D4×#1853, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287272);