Finite Group Databasebeta

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Cayley diagram

SD16×#131 #6345799

order512
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productSD16×#131

Subgroups

subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalD4×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C4×#131 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

1840 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);(2,4)(3,7)(6,8);(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

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSD16×#444, #72321893, #72342636, …
maximal normal subgroup ofSD16×#444, #72321893, #72342636, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252995);