Finite Group Databasebeta

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Cayley diagram

C2×#26800 #6443762

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

Aliases

direct productC2×#26800

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2592 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);(4,5)(7,8)(9,14)(10,12)(11,13);(3,4,6,5)(7,9,11,10)(8,12,13,14)(16,19)(17,21)(20,22);(3,4,6,5)(7,10,11,9)(8,12,13,14)(15,16)(17,22)(18,20)(19,21);(3,5,6,4)(8,13)(12,14)(16,20)(19,22)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6576038, C2×#6577350, C2×#6444783
maximal normal subgroup ofC2×#6576038, C2×#6577350, C2×#6444783

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350958);