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#72306754

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#512862
commutator, Frattini, cosocle#493C24
prime radical#512867C2
LCSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G
LECSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1387 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1387 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=ghij,d^2=j,e^2=h,g^2=i,h^2=j,b^a=bij,c^a=ci,c^b=cij,d^a=de,d^b=df,d^c=dj,e^a=ej,e^b=eg,e^c=ej,e^d=ehj,f^a=fghj,f^c=fj,f^e=fh,g^b=gi,g^d=gh,g^f=gi,h^b=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306759
maximal normal subgroup of#72306759

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