Finite Group Databasebeta

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Cayley diagram

C23×#360 #7622504

order512
nilpotency class4
isoclinismisoclinic class of #360
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC23×#360

Subgroups

subgroupG/H
center, socleC24#102
commutatorC8C24×C4
Frattini, cosocleC2×C8C25
prime radicalC23×#135C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

4258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#148057, C23×AGL(1,ℤ/16), C2×#147939, …
maximal normal subgroup ofC2×#148057, C23×AGL(1,ℤ/16), C2×#147939, …
maximal quotient ofC2×#495527, C22×#13001, #72265205, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529700);