Finite Group Databasebeta

← Prev (#72285972) Next (#72285974) →

#72285973

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151462
commutator, Frattini, cosocle#1781C23
prime radical#10336C22
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

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

normal subgroups only

This group has 515 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:

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

Polycyclic presentation:

2^10|a^2=d,c^2=gh,d^2=j,e^2=g,b^a=be,c^a=cdfij,c^b=cefghij,d^b=df,d^c=dh,e^a=efg,e^b=eg,e^c=eghi,e^d=ehij,f^a=fh,f^c=fhij,f^e=fij,g^a=gij,g^c=gj,g^d=gj,h^b=hi,h^c=hj,h^e=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285989, #72286122
maximal normal subgroup of#72285989, #72286122

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