Finite Group Databasebeta

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#72310971

order1024
nilpotency class3
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C24×C43

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#46996
commutator, Frattini, cosocleC25C25
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

3781 conjugacy classes of subgroups; only the 415 normal subgroups are shown.

normal subgroups only

This group has 3781 conjugacy classes of subgroups — too many to draw. Only its 415 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 — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=i,d^2=ghj,b^a=bh,c^b=cf,d^a=dfg,d^b=dfij,d^c=dgh,e^a=ehij,e^b=ehj,e^c=eghi,e^d=ei,f^a=fi,f^d=fij,f^e=fj,g^a=gi,g^b=gij,g^c=gi,g^d=gj,h^c=hi,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72311029, #72337705, #72337708, …
maximal normal subgroup of#72311029, #72337705, #72337708, …

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