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Cayley diagram

C16×H(5,𝔽2) #7622756

order512
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphism?
homologyAbelianization (H1) = C24×C16 · Schur multiplier (H2) = C29

Aliases

direct productC16×H(5,𝔽2)

Subgroups

subgroupG/H
centerC2×C16C24
commutatorC2C24×C16
Frattini, cosocleC2×C8C25
socleC22C24×C8
prime radicalC2×H(5,𝔽2)C8
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C16 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

1592 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:

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

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup ofC16×#997, C16×#5935
maximal subgroup ofC16×B(4,𝔽2), H(5,𝔽2)×AGL(1,𝔽17), C16×#453, …
maximal normal subgroup ofC16×B(4,𝔽2), C16×#453, C16×#997, …
maximal quotient ofH(5,𝔽2)×AGL(1,𝔽17)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529952);