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Cayley diagram

C2×C8×H(5,𝔽2) #10574463

order512
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphism?
homologyAbelianization (H1) = C25×C8 · Schur multiplier (H2) = C214

Aliases

direct productC2×C8×H(5,𝔽2)

Subgroups

subgroupG/H
centerC22×C8C24
commutatorC2C25×C8
Frattini, cosocleC2×C4C26
socleC23C24×C4
prime radicalC22×H(5,𝔽2)C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

9098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 18 points:

(1,2);(3,4,5,6,7,8,9,10);(11,12)(13,17)(14,15)(16,18);(12,15)(16,17);(11,12)(13,16)(14,15)(17,18);(11,13)(12,16)(14,18)(15,17)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC2×#6344533, C2×#10574511, #106×H(5,𝔽2), …
maximal normal subgroup ofC2×#6344533, C2×#10574511, #106×H(5,𝔽2), …
maximal quotient ofC2×C8×D42, #72315511, #72322106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481659);