Finite Group Databasebeta

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#72284342

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#142568
commutator, Frattini, cosocle#2309C23
prime radical#251828C2
LCSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G
LECSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1415 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 1415 conjugacy classes of subgroups — too many to draw. Only its 59 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=ghj,d^2=ghj,e^2=ghj,b^a=bfhj,c^a=cd,c^b=ce,d^a=dg,d^b=dij,d^c=dgh,e^a=ehi,e^b=eghj,e^c=eghj,e^d=ej,f^b=fij,f^c=fhj,f^d=fi,f^e=fij,g^a=gi,g^b=gij,g^d=gj,g^e=gj,h^a=hij,h^b=hij,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284479, #72285651, #72285854, …
maximal normal subgroup of#72284479, #72285651, #72285854, …

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