Finite Group Databasebeta

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Cayley diagram

C4×#2669 #521168

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4

Aliases

direct productC4×#2669

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#2669C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

5806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 16 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#2669, #72280965, #72280997, …
maximal normal subgroup of#72280965, #72280997, #72281112, …
maximal quotient ofAGL(1,𝔽5)×#2669, C4×#15258, C4×#15310, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428364);