Finite Group Databasebeta

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#72238434

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#8534
commutatorC23×D4C22×C4
Frattini, cosocleC22×#123C23
prime radical#15947C4
LCSG ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G
LECSG ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1

2429 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 2429 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(3,13)(4,18)(9,16)(10,24)(17,25)(20,22)(21,29)(23,27);(1,2,6,5)(3,14,13,30)(4,10,18,24)(7,11,8,12)(9,26,16,19)(15,27,31,23)(17,28,25,32)(20,21,22,29);(1,3,10,28,7,13,29,32)(2,9,20,15,11,16,18,31)(4,19,5,23,22,26,12,27)(6,25,24,30,8,17,21,14)

Polycyclic presentation:

2^10|b^2=h,c^2=dfj,d^2=j,f^2=j,b^a=bgij,c^a=ce,c^b=cd,d^a=dh,d^c=dg,e^b=ehij,e^c=eh,e^d=ej,f^b=fgj,f^c=fg,f^d=fi,g^c=gi,h^c=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238436, #72238666
maximal normal subgroup of#72238436, #72238666

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.4*F.6*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);