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Cayley diagram

D4×#471 #6574091

order512
nilpotency class3
isoclinismisoclinic class of D4×#471
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productD4×#471

Subgroups

subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalD4×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2634 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);(6,10)(7,13)(8,16)(12,18)(14,15)(17,19);(5,6,8,10,9,12,16,18)(7,11,14,17,15,20,13,19);(5,7)(6,11)(8,14)(9,15)(10,17)(12,20)(13,16)(18,19)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72322841, #72322874

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481287);