Finite Group Databasebeta

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#72283471

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#32898
commutator, Frattini, cosocleC22×C22⋊C4C24
prime radical#8932883C2
LCSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

5138 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 5138 conjugacy classes of subgroups — too many to draw. Only its 199 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=i,c^2=h,d^2=ei,f^2=ij,g^2=i,c^a=ch,c^b=ci,d^a=de,d^b=dg,d^c=df,e^b=ei,e^c=eij,f^a=fi,f^b=fh,f^c=fi,g^a=gi,g^b=gi,g^c=ghj,g^f=gj,h^d=hj

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283556, #72283720
maximal normal subgroup of#72283556, #72283720
maximal quotient of#72284154, #72284448

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