Finite Group Databasebeta

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#72283782

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151342
commutator, Frattini, cosocle#1781C23
prime radical#151130C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

722 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 37 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:

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

Polycyclic presentation:

2^10|a^2=fghi,b^2=f,c^2=defhj,e^2=gij,b^a=bhi,c^a=cd,c^b=cdegij,d^a=dh,d^b=df,d^c=dfhj,e^a=efij,e^b=efg,e^c=ef,e^d=ehij,f^a=fi,f^c=fh,f^e=fij,g^a=gij,g^b=gij,g^c=gi,g^d=gj,h^a=hj,h^b=hi,h^c=hij,h^e=hj,i^c=ij

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283793, #72283923, #72285770, …
maximal normal subgroup of#72283793, #72283923, #72285770, …

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