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Cayley diagram

D4×#472 #6574092

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

Aliases

direct productD4×#472

Subgroups

subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2755 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)(7,14)(8,11)(13,18)(15,16)(17,19);(5,6,11,18,9,13,8,10)(7,12,14,19,16,20,15,17);(5,7)(6,12)(8,15)(9,16)(10,17)(11,14)(13,20)(18,19)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72322937, #72323014

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481288);