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#72267851

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#12090
commutator, Frattini, cosocle#1445C23
prime radical#3492D4
LCSG ▷ #1445 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1445 ◁ G
LECSG ▷ #1445 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1

746 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 64
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,10)(3,7,15,25)(4,16,18,8)(5,9,11,21)(12,29,31,26)(13,23,14,24)(17,32,30,22)(19,27,20,28)(34,39)(35,43)(36,47)(40,56)(41,59)(42,50)(44,62)(45,64)(46,48)(49,52)(51,61)(54,57)(55,60)(58,63);(1,3,11,13,6,15,5,14)(2,7,21,23,10,25,9,24)(4,17,19,12,18,30,20,31)(8,26,27,22,16,29,28,32)(33,34)(35,44,46,40)(36,41)(37,50,53,39)(38,42)(43,57,48,63)(45,54)(47,60)(49,59,52,55)(51,58)(56,64)(61,62);(1,4,11,20,6,18,5,19)(2,8,21,28,10,16,9,27)(3,12,13,17,15,31,14,30)(7,22,23,26,25,32,24,29)(33,35,38,46)(34,40,42,44)(36,48,47,43)(37,51)(39,54)(41,57,60,63)(45,53)(49,61)(50,58)(52,64)(55,56)(59,62)

Polycyclic presentation:

2^10|a^2=i,b^2=fi,c^2=fgij,d^2=j,e^2=g,f^2=i,b^a=be,c^a=cd,c^b=cfgh,d^a=dj,d^b=df,d^c=dhj,e^a=eg,e^b=egh,e^c=efj,e^d=ehj,f^a=fh,f^c=fij,f^d=fi,f^e=fj,g^c=gij,h^b=hj,h^c=hi

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72267855

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