Finite Group Databasebeta

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#72306527

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

1440 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 1440 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:

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

Polycyclic presentation:

2^10|a^2=gj,g^2=ij,h^2=j,b^a=bij,c^a=cf,c^b=ce,d^a=dh,d^b=dghj,e^a=eg,e^d=ehi,f^a=fhj,f^b=fghij,f^e=fhi,g^c=gh,g^d=gij,g^e=gij,g^f=gij,h^a=hi,h^b=hi,h^c=hj,h^d=hj,h^e=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306535, #72306540, #72306556, …
maximal normal subgroup of#72306535, #72306540, #72306556, …

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