Finite Group Databasebeta

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#72283355

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

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

1976 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1976 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) = 16
transitive degreeμt(G) = 16

A minimal faithful permutation representation — 3 generators on 16 points:

(3,9)(6,13)(7,14)(10,16)(12,15);(1,2)(3,10,13,15)(4,14)(5,8)(6,12)(7,11)(9,16);(1,3,11,16)(2,6,7,15,8,9,14,10)(4,12,5,13)

Polycyclic presentation:

2^10|b^2=fghi,c^2=dfj,d^2=gh,e^2=g,b^a=bgi,c^a=ce,c^b=cd,d^a=df,d^b=dg,d^c=dh,e^a=eg,e^b=efhi,e^c=efgij,e^d=ehj,f^b=fj,f^c=fh,f^d=fi,f^e=fij,g^b=gi,g^c=gij,g^d=gj,h^a=hi,h^b=hij,h^c=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283462, #72283793, #72285692, …
maximal normal subgroup of#72283462, #72283793, #72285692, …

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