Finite Group Databasebeta

← Prev (#6573665) Next (#6573667) →

Cayley diagram

D4×#456 #6573666

order512
nilpotency class3
isoclinismisoclinic class of D4×#453
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productD4×#456

Subgroups

subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalD4×2−1+4C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4548 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);(2,4);(5,6,10,8)(7,13,16,20)(9,17,19,12)(11,18,14,15);(5,7)(6,11)(8,14)(9,18)(10,16)(12,20)(13,17)(15,19);(5,8,10,6)(7,14,16,11)(9,12,19,17)(13,18,20,15)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4×#3018, #72312395, #72313645
maximal normal subgroup ofD4×#3018, #72312395, #72313645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480862);