Finite Group Databasebeta

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#72314644

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#251827
commutator, Frattini, cosocleC23×D4C24
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

2571 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 2571 conjugacy classes of subgroups — too many to draw. Only its 131 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)(3,7)(4,14)(5,17,19,9)(6,10)(8,15)(11,20,26,30)(12,27)(13,22)(16,23,18,24)(21,29)(25,32,28,31);(1,3)(2,7)(4,12)(5,11)(6,13)(8,21)(9,20)(10,22)(14,27)(15,29)(16,25)(17,30)(18,28)(19,26)(23,31)(24,32);(3,11)(4,15)(7,20)(8,14)(12,28)(13,26)(16,18)(21,32)(22,30)(23,24)(25,29)(27,31);(1,4,6,15)(2,8,10,14)(3,12,13,29)(5,18,19,16)(7,21,22,27)(9,24,17,23)(11,25,26,28)(20,31,30,32)

Polycyclic presentation:

2^10|a^2=f,d^2=j,b^a=bf,c^a=cg,c^b=ce,d^a=dj,d^b=df,d^c=dj,e^a=eh,e^d=ehi,f^c=fhij,f^e=fj,g^a=ghij,g^b=gi,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72314648, #72314667, #72314678
maximal normal subgroup of#72314648, #72314667, #72314678

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