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#72227652

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2×C4#2199
commutator, Frattini, cosocle#2285C23
socleC22#13343
prime radicalC24B(4,𝔽2)
LCSG ▷ #2285 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #2285 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

945 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 945 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,f^2=j,h^2=j,b^a=bd,c^a=ce,c^b=cf,d^a=dg,d^b=dg,d^c=dh,e^a=eg,e^b=ehij,e^c=eg,e^d=eij,f^b=fj,f^c=fj,h^a=hi,h^c=hj,h^d=hj,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227661
maximal quotient of#72227656

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