Finite Group Databasebeta

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#72285573

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151366
commutator, Frattini, cosocle#1781C23
prime radical#245290C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1000 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 39 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:

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

Polycyclic presentation:

2^10|b^2=ij,c^2=ghj,d^2=g,e^2=gh,b^a=bj,c^a=cd,c^b=ce,d^a=dg,d^b=df,d^c=dgi,e^a=efhj,e^b=eghj,e^c=egh,e^d=ehj,f^a=fi,f^c=fh,f^d=fi,f^e=fij,g^b=gi,g^e=gj,h^a=hi,h^b=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285603, #72286199, #72340929, …
maximal normal subgroup of#72285603, #72286199, #72340929, …

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