Finite Group Databasebeta

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#72251740

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#144627
commutator, Frattini, cosocleC2×#535C23
prime radical#52004C22
LCSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G
LECSG ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

986 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 53 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:

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

Polycyclic presentation:

2^10|a^2=gij,b^2=ef,c^2=defh,d^2=i,h^2=j,b^a=be,c^a=cd,c^b=cfj,d^a=dij,d^b=dh,d^c=dg,e^b=egij,e^c=eghi,e^d=ei,f^a=fgij,f^b=fgij,f^c=fhj,f^d=fij,g^b=gj,g^c=gj,h^a=hi,h^c=hij,h^e=hj,h^f=hj,i^b=ij

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72251744, #72251761, #72251766
maximal normal subgroup of#72251744, #72251761, #72251766

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