Finite Group Databasebeta

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#72240083

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#10390
commutatorC23×D4C22×C4
Frattini, cosocleC24×D4C23
prime radicalC23×#127C22
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C24×D4 ◁ G
LECSG ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1

4237 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 4237 conjugacy classes of subgroups — too many to draw. Only its 141 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) = 32

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=d,c^2=h,b^a=bf,c^a=cdghj,c^b=cde,d^a=dh,d^c=dh,e^a=ehi,e^b=ehj,e^c=eh,e^d=ej,f^a=fhij,f^b=fh,f^c=fh,f^d=fj,g^b=ghij,g^c=gh,g^d=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240084, #72255894, #72255907, …
maximal normal subgroup of#72240084, #72255894, #72255907, …
maximal quotient of#72240090

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