Finite Group Databasebeta

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#72283389

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#13561
commutatorC2×#127C22×C4
Frattini, cosocleC2×#492C23
prime radicalC2×#3151C22
LCSG ▷ C2×#127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C22×C42 ◁ C2×#3151 ◁ G
LECSG ▷ C2×#492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1907 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 1907 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 36
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(2,4)(6,12)(7,17)(8,22)(15,16)(19,25)(21,26)(30,32)(33,35);(5,6)(7,18,26,27,17,31,21,13)(8,23)(9,15)(10,16)(11,12)(14,25)(19,28)(20,33,36,32,24,35,34,30)(22,29);(1,2,3,4)(5,7,19,34,10,26,22,20,11,17,25,36,9,21,8,24)(6,13,29,35,15,18,28,30,12,27,23,33,16,31,14,32)

Polycyclic presentation:

2^10|b^2=ghij,c^2=dehij,e^2=hi,g^2=i,h^2=j,b^a=bj,c^a=cd,c^b=cf,d^c=dj,e^a=ej,e^b=eg,e^c=ej,f^b=fhj,f^c=fg,f^e=fhij,g^b=gi,g^c=gh,g^f=gi,h^b=hi,h^c=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283469, #72283567, #72283945
maximal normal subgroup of#72283469, #72283567, #72283945

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