Finite Group Databasebeta

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Cayley diagram

C2×#26647 #6439519

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

Aliases

direct productC2×#26647

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2428C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2916 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)(9,12)(10,11)(13,14)(15,16,20,17,19,22,18,21);(3,4)(5,6)(15,17,19,21)(16,18,22,20);(4,6)(8,11)(12,14)(15,18,19,20)(16,17,22,21);(7,9,10,13)(8,12,11,14)(15,16)(17,18)(19,22)(20,21)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7515961, C2×#6439861, C2×#7521930
maximal normal subgroup ofC2×#7515961, C2×#6439861, C2×#7521930

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346715);