Finite Group Databasebeta

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Cayley diagram

C2×#23521 #6431924

order512
nilpotency class3
isoclinismisoclinic class of #23439
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23521

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(3,4)(5,6)(7,8)(9,16)(10,12)(11,14)(13,17)(15,19)(18,21)(20,22);(8,12)(9,17)(10,19)(13,21)(14,15)(16,20);(4,6)(7,9,10,18,11,17,19,22)(8,13,15,21,14,16,12,20);(8,14)(12,15)(13,16)(20,21)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7500757, #72237173, #72237176, …
maximal normal subgroup ofC2×#7500757, #72237173, #72237176, …
maximal quotient of#72238959, #72239014, D3×#23521, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339120);