Finite Group Databasebeta

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Cayley diagram

SD16×#123 #6574812

order512
nilpotency class3
isoclinismisoclinic class of D8×#120
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productSD16×#123

Subgroups

subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#123 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2,3,4,5,6,7,8);(2,4)(3,7)(6,8);(10,12)(14,17)(15,19);(13,14)(15,18)(16,17)(19,20);(9,10)(11,12)(13,15,16,19)(14,18,17,20)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72305389, SD16×#525, #123×AGL(1,ℤ/8)
maximal normal subgroup of#72305389, SD16×#525, #123×AGL(1,ℤ/8)
maximal quotient of#72305866, #72306025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482008);