Finite Group Databasebeta

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#72314650

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

2309 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 2309 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)(3,11)(4,15)(5,9,20,25)(6,10)(7,14)(8,18)(12,27,29,21)(13,22)(16,23)(17,26,19,24)(28,31,30,32);(1,3)(2,7)(4,16)(5,12)(6,14)(8,23)(9,21)(10,11)(13,18)(15,22)(17,30)(19,28)(20,29)(24,32)(25,27)(26,31);(3,12)(7,21,11,27)(9,25)(13,28)(14,29)(16,30)(22,31,23,32)(24,26);(1,4,6,18)(2,8,10,15)(3,13,14,16)(5,19,20,17)(7,22,11,23)(9,26,25,24)(12,28,29,30)(21,31,27,32)

Polycyclic presentation:

2^10|a^2=f,c^2=g,d^2=j,b^a=bj,c^a=cfg,c^b=ce,d^a=dj,d^b=dfj,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

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72314655, #72314663, #72314678
maximal normal subgroup of#72314655, #72314663, #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, 3, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*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*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);