Finite Group Databasebeta

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#72285117

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
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

1214 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1214 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:

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

Polycyclic presentation:

2^10|a^2=fghj,b^2=dgij,c^2=fghi,d^2=fhj,e^2=f,b^a=bd,c^a=cj,c^b=ce,d^a=df,d^b=dhj,d^c=dg,e^a=eghij,e^b=efgij,e^c=efhj,e^d=ehj,f^a=fij,f^b=fi,f^c=fj,f^d=fj,g^a=gj,g^b=gh,g^c=gij,g^d=gij,g^e=gi,h^a=hi,h^b=hij,h^c=hi,h^d=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72285153, #72285637, #72286141, …
maximal normal subgroup of#72285153, #72285637, #72286141, …

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