Finite Group Databasebeta

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#72318310

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139816
commutatorC2×#127C22×C4
Frattini, prime radical, cosocleC2×#535C23
LCSG ▷ C2×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

905 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 905 conjugacy classes of subgroups — too many to draw. Only its 69 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:

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

Polycyclic presentation:

2^10|a^2=d,b^2=degij,c^2=fij,h^2=j,b^a=be,c^a=cef,c^b=cegh,d^b=dg,d^c=dh,e^a=eg,e^b=egj,e^c=egh,f^a=fghi,f^b=fghi,f^c=fj,f^d=fi,f^e=fij,g^c=gj,h^a=hi,h^b=hj,h^c=hij,h^d=hj,h^e=hj,h^f=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72318311, #72318472, #72318857, …
maximal normal subgroup of#72318311, #72318472, #72318857, …

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