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#72245391

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#500179
commutator, Frattini, cosocle#532C24
prime radical#7487865C2
LCSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #3444 ◁ G
LECSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1544 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1544 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ehi,d^2=h,g^2=i,h^2=j,b^a=bf,c^a=ce,c^b=cj,d^a=di,d^b=dg,d^c=dij,e^b=ei,e^d=ej,f^a=fi,f^b=fj,f^c=fij,f^d=fh,f^e=fj,g^a=ghij,g^b=gij,g^f=gi,h^b=hi,h^f=hj,i^a=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245425
maximal normal subgroup of#72245425

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