Finite Group Databasebeta

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#72258405

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#10336
commutatorC23×D4C22×C4
Frattini, cosocleC24×D4C23
prime radicalC2×#23895C2
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C24×D4 ◁ G
LECSG ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1

5003 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 5003 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,3,4)(6,10)(7,13)(8,17)(12,18)(14,19)(15,16);(5,6)(7,14)(8,18)(9,12)(10,17)(11,13)(15,20)(16,19);(5,7,9,16)(6,11,12,20)(8,13,17,15)(10,19,18,14)

Polycyclic presentation:

2^10|a^2=g,c^2=j,d^2=j,b^a=bd,c^a=cdf,c^b=cde,d^a=dj,d^b=dj,d^c=dh,e^a=ei,e^b=ej,e^c=eh,e^d=ej,f^b=fhi,f^c=fhj,h^b=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258411, #72258443, #72258465
maximal normal subgroup of#72258411, #72258443, #72258465
maximal quotient of#72312430, #72337737

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