Finite Group Databasebeta

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Cayley diagram

C2×#23961 #6451789

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23961

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5218 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)(10,12)(11,14)(16,20)(17,18)(21,22);(7,9,10,14)(8,11,12,13)(16,21)(17,18)(20,22);(4,6)(15,16,19,22)(17,21,18,20);(8,12)(11,13)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6453899, C2×#6453971, C2×#6451823
maximal normal subgroup ofC2×#6453899, C2×#6453971, C2×#6451823

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358985);