Finite Group Databasebeta

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Cayley diagram

D8×C22⋊Q8 #6574434

order512
nilpotency class3
isoclinismisoclinic class of D8×#120
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productD8×C22⋊Q8

Subgroups

subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2568 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);(2,8)(3,7)(4,6);(10,12)(13,14,16,18)(15,19,20,17);(13,15,16,20)(14,17,18,19);(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282092, #72305388, C22⋊Q8×AGL(1,ℤ/8), …
maximal normal subgroup of#72282092, #72305388, C22⋊Q8×AGL(1,ℤ/8), …
maximal quotient of#72305837, #72319556, #72322084, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481630);