Finite Group Databasebeta

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#72306896

order1024
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#512862
commutator, Frattini, cosocle#493C24
LCSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3151 ◁ G
LECSG ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

1401 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1401 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=gj,b^2=gj,d^2=j,f^2=hj,g^2=ij,h^2=j,b^a=bij,c^a=ci,d^a=df,d^b=de,d^c=dj,e^a=eg,e^b=ehj,f^b=fghj,f^c=fj,f^d=fh,f^e=fh,g^d=gh,g^e=gij,h^b=hi,h^d=hj,h^e=hj,i^d=ij

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306902, #72338309, #72338347
maximal normal subgroup of#72306902, #72338309, #72338347

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