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#72285339

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#485605
commutator, Frattini, cosocle#381C24
LCSG ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2280 ◁ G
LECSG ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1668 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1668 conjugacy classes of subgroups — too many to draw. Only its 121 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)(3,12)(4,5)(6,21)(7,11)(8,16)(9,20)(10,24)(13,28)(14,17)(15,27)(18,19)(22,26)(23,32)(25,29)(30,31);(1,3,7,16)(2,8,11,12)(4,17)(5,14)(6,22,24,15)(9,25)(10,26,21,27)(13,23)(18,30)(19,31)(20,29)(28,32);(1,4,5,2)(3,13,29,26,16,30,14,27)(6,23,19,10)(7,9,20,11)(8,22,25,28,12,15,17,31)(18,32,21,24);(1,5)(2,9,11,4)(3,14,16,29)(6,19)(7,20)(8,25)(10,18,21,23)(12,17)(13,27)(15,28,22,31)(24,32)(26,30)

Polycyclic presentation:

2^10|b^2=ghj,c^2=eh,d^2=hi,e^2=h,f^2=j,b^a=bghj,c^a=ce,c^b=cf,d^a=dg,d^b=dg,d^c=dgh,e^a=eh,e^b=eg,e^d=ei,f^a=fgj,f^b=fij,f^c=fg,f^d=fi,f^e=fi,g^b=gj,g^c=gi,g^d=gj,g^e=gj,h^b=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285435
maximal normal subgroup of#72285435

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