Finite Group Databasebeta

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#72292057

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#12090
commutator, Frattini, cosocleC2×#379C23
LCSG ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C2×#379 ◁ G
LECSG ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

3175 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 3175 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) = 20
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 20 points:

(6,10)(11,17)(12,18)(13,15)(19,20);(2,4)(5,6,9,12)(7,11,14,19)(8,10,16,18)(13,20,15,17);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,16)(18,20)

Polycyclic presentation:

2^10|b^2=j,c^2=d,f^2=h,b^a=bf,c^a=ce,c^b=cd,e^b=eg,f^a=fh,f^b=fh,f^c=fgi,f^e=fij,g^a=gi,g^f=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72292059, #72292076, #72292080
maximal normal subgroup of#72292059, #72292076, #72292080

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