Finite Group Databasebeta

← Prev (#6439038) Next (#6439040) →

Cayley diagram

C2×#26368 #6439039

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

Aliases

direct productC2×#26368

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2894 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);(7,8,11,13)(9,12,10,14)(15,16)(17,19)(18,21)(20,22);(3,4)(5,6)(7,9,11,10)(8,12,13,14)(16,19)(17,22)(20,21);(4,6)(7,10,11,9)(8,12,13,14);(16,19)(17,22)(20,21)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6439784, C2×D8×C22⋊Q8, C2×#6577186
maximal normal subgroup ofC2×#6439784, C2×D8×C22⋊Q8, C2×#6577186

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346235);