Finite Group Databasebeta

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Cayley diagram

C2×#24186 #6448785

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24186

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264445, #72264485, #72264571, …
maximal normal subgroup of#72264445, #72264485, #72264571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355981);