Finite Group Databasebeta

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#72284064

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151391
commutator, Frattini, cosocle#1781C23
prime radicalC23×D4C2×D4
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

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

normal subgroups only

This group has 452 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)(3,8,16,32)(4,17)(5,10,24,38)(6,25,28,11)(7,12)(9,19)(13,41)(14,36)(15,31)(18,52,47,49)(20,50,27,59)(21,51)(22,35)(23,37)(26,39)(29,44)(30,46)(33,53,62,56)(34,55,40,54)(42,60)(43,64)(45,61,48,63)(57,58);(1,3,7,16)(2,8,12,32)(4,18,21,47)(5,14,24,46)(6,26)(9,33,35,62)(10,36,38,30)(11,39)(13,27)(15,28)(17,49)(19,53)(20,43)(22,56)(23,45)(25,31)(29,40)(34,60)(37,63)(41,50,64,59)(42,55,44,54)(48,57)(51,52)(58,61);(1,4,7,21)(2,9)(3,13,16,43)(5,22,24,19)(6,27)(8,29)(10,17)(11,40,25,34)(12,35)(14,44,46,42)(15,47)(18,26)(20,28)(23,54)(30,41)(31,62,39,33)(32,60)(36,64)(37,59,58,50)(38,51)(45,53)(48,56)(49,63,52,61)(55,57)

Polycyclic presentation:

2^10|a^2=fg,b^2=fghj,c^2=ghj,d^2=gh,b^a=bghj,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dghj,e^a=efhi,e^b=eh,e^c=ei,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

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72284081, #72285870
maximal normal subgroup of#72284081, #72285870

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