Finite Group Databasebeta

← Prev (#6438195) Next (#6438197) →

Cayley diagram

C2×#24916 #6438196

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#24916

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#449C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3210 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);(7,8)(9,11)(10,12)(13,14)(16,21)(17,19)(18,22);(3,4)(5,6)(7,9,10,13)(8,11,12,14)(15,16)(17,21)(18,20)(19,22);(4,6)(8,12)(11,14)(15,17,20,19)(16,21,18,22);(15,18,20,16)(17,22,19,21)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7517063, C2×#6439861, C2×#7520384
maximal normal subgroup ofC2×#7517063, C2×#6439861, C2×#7520384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345392);