Finite Group Databasebeta

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#72339743

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#18992
commutator, Frattini, cosocle#376C24
socleC2#166100
prime radical#233608C2
LCSG ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1

1540 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1540 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,12,15,27,17,29,16,32)(4,18)(5,21,23,20)(6,11)(7,9)(8,14,25,22,24,30,19,28)(10,13,26,31);(1,3,13,30,7,17,31,14)(2,8,20,29)(4,19,26,32)(5,22,6,16,23,28,18,15)(9,24,21,12)(10,27,11,25);(1,4)(2,6)(3,8)(5,10)(7,11)(9,18)(12,28)(13,21)(14,32)(15,19)(16,25)(17,24)(20,31)(22,29)(23,26)(27,30);(2,9)(4,11)(5,23)(8,24)(13,31)(14,30)(19,25)(22,28)

Polycyclic presentation:

2^10|a^2=f,b^2=efhij,e^2=gh,f^2=i,g^2=j,h^2=j,b^a=be,c^a=cfij,c^b=cfi,d^a=dij,d^b=dij,d^c=dj,e^a=eg,e^b=eh,e^c=ehj,e^d=ej,f^b=fh,f^c=fi,f^e=fj,g^b=gj,h^a=hj,h^c=hj,i^b=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72339744, #72339808
maximal normal subgroup of#72339744, #72339808

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