Finite Group Databasebeta

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#72339017

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#139472
commutatorC2×#117C22×C4
Frattini, cosocle#2309C23
prime radical#422772C2
LCSG ▷ C2×#117 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #35272 ◁ G
LECSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1353 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1353 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=d,c^2=fhj,d^2=g,f^2=j,b^a=bg,c^a=cf,c^b=cde,d^c=dh,e^a=eg,e^b=eghij,e^c=ehj,e^d=eij,f^b=fg,f^c=fj,f^e=fj,g^c=gj,h^b=hi,h^c=hj,h^d=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72339018, #72339023, #72339045
maximal normal subgroup of#72339018, #72339023, #72339045

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