Finite Group Databasebeta

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#72284047

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151196
commutator, Frattini, cosocle#1781C23
prime radical#23895C22
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

690 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 64

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

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

Polycyclic presentation:

2^10|a^2=fgj,b^2=fghj,c^2=h,d^2=gh,e^2=j,b^a=bgh,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dghi,e^a=efhi,e^b=ehj,e^d=ehij,f^a=fij,f^b=fi,f^c=fh,f^d=fi,g^a=gij,g^b=gij,g^d=gj,g^e=gj,h^a=hi,h^b=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284072, #72284775, #72284923, …
maximal normal subgroup of#72284072, #72284775, #72284923, …

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