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Cayley diagram

M16×H(5,𝔽2) #10574527

order512
nilpotency class2
isoclinismisoclinic class of D4×H(5,𝔽2)
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productM16×H(5,𝔽2)

Subgroups

subgroupG/H
center, Frattini, cosocleC2×C4C26
commutator, socleC22C25×C4
prime radicalC22×H(5,𝔽2)C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

7864 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);(2,6)(4,8);(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

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,ℤ/8)×H(5,𝔽2), #72267603, #72313064, …
maximal normal subgroup ofAGL(1,ℤ/8)×H(5,𝔽2), #72267603, #72313064, …
maximal quotient of#72257041, #72264731, #72283704, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481723);