Finite Group Databasebeta

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#72285849

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#151130
commutator, Frattini, cosocle#1781C23
prime radical#144378C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1335 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1335 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:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=efghi,e^2=gh,b^a=bfi,c^a=cd,c^b=ce,d^b=df,d^c=dfj,e^a=efi,e^b=eg,e^c=eh,e^d=ehj,f^c=fh,f^e=fij,g^a=gij,g^b=gij,g^c=gi,g^d=gj,g^e=gj,h^b=hi,h^c=hij,h^e=hj,i^c=ij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285860, #72286031, #72310913
maximal normal subgroup of#72285860, #72286031, #72310913

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