Finite Group Databasebeta

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#72314643

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

2281 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 2281 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)(5,10,18,23)(6,8)(9,15)(11,21,27,19)(12,28)(13,22)(16,25,17,24)(20,30)(26,31,29,32);(1,3)(2,7)(4,12)(5,11)(6,13)(8,22)(9,20)(10,21)(14,28)(15,30)(16,26)(17,29)(18,27)(19,23)(24,32)(25,31);(2,8)(3,11)(4,15)(7,19,22,21)(12,29)(13,27)(16,17)(20,32,28,31)(24,25)(26,30);(1,4,6,15)(2,9,8,14)(3,12,13,30)(5,17,18,16)(7,20,22,28)(10,25,23,24)(11,26,27,29)(19,31,21,32)

Polycyclic presentation:

2^10|a^2=f,c^2=gj,d^2=j,c^a=cfg,c^b=ce,d^a=dj,d^b=df,d^c=dj,e^a=ei,e^c=eij,e^d=eh,f^c=fhj,f^e=fj,g^a=ghj,g^b=gij,h^b=hj,i^a=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72314648, #72314663, #72314682
maximal normal subgroup of#72314648, #72314663, #72314682

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