Finite Group Databasebeta

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Cayley diagram

M16×#131 #6342515

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) order 196608
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4

Aliases

direct productM16×#131

Subgroups

subgroupG/H
centerC4×C8C24
commutator, socleC22C23×C42
Frattini, cosocleC42C25
prime radicalC22×#41C2×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C4×C8 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

2271 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2,3,4,5,6,7,8);(2,6)(4,8);(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#131×AGL(1,ℤ/8), #72342644, M16×#444
maximal normal subgroup of#131×AGL(1,ℤ/8), #72342644, M16×#444

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249711);