Finite Group Databasebeta

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#72228009

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#13088
commutatorC22×C42C22×C4
Frattini, cosocleC2×#532C23
prime radical#35513C22
LCSG ▷ C22×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #35513 ◁ G
LECSG ▷ C2×#532 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

1451 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 1451 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 128

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

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

Polycyclic presentation:

2^10|b^2=e,c^2=fg,d^2=i,e^2=j,g^2=j,b^a=bf,c^a=cf,c^b=cd,d^a=dh,d^b=dg,d^c=dhj,e^a=ehj,e^c=egij,e^d=eij,f^b=fhj,f^c=fj,f^e=fj,g^a=gj,g^b=gi,g^c=gj,g^e=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228015
maximal normal subgroup of#72228015

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.5);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);