Finite Group Databasebeta

← Prev (#72285784) Next (#72285786) →

#72285785

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#142568
commutator, Frattini, cosocle#2309C23
prime radical#251855C2
LCSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G
LECSG ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1

1408 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 59 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)(5,18)(7,21)(8,23)(9,19)(12,25)(13,29)(14,15)(16,30)(20,26)(22,32)(27,28);(1,2)(3,12)(4,8)(5,19,18,9)(6,10)(7,13)(11,25,15,27)(14,28)(16,24,30,31)(17,23)(20,22)(21,29,32,26);(1,3)(2,7,10,22)(4,15,17,11)(5,20,18,13)(6,14)(8,21)(9,12)(16,26)(19,28)(23,32)(24,27,31,25)(29,30)

Polycyclic presentation:

2^10|b^2=fhij,c^2=ghj,d^2=g,e^2=gj,b^a=bf,c^a=cd,c^b=ce,d^a=dg,d^b=dh,d^c=dgi,e^a=ej,e^b=egh,e^c=egi,f^b=fi,f^c=fh,f^d=fij,f^e=fi,g^b=gj,h^a=hi,h^b=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285786, #72285910, #72338963
maximal normal subgroup of#72285786, #72285910, #72338963

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