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Cayley diagram

D8×H(5,𝔽2) #10583013

order512
nilpotency class3
isoclinismisoclinic class of D8×H(5,𝔽2)
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C214

Aliases

direct productD8×H(5,𝔽2)

Subgroups

subgroupG/H
center, socleC22C24×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×H(5,𝔽2) ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

9134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 6 generators on 16 points:

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267592, AGL(1,ℤ/8)×H(5,𝔽2), #72267616, …
maximal normal subgroup of#72267592, AGL(1,ℤ/8)×H(5,𝔽2), #72267616, …
maximal quotient of#72283117, D42×D8, #102×H(5,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10490209);