Finite Group Databasebeta

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#72238057

order1024
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#12187
commutator, Frattini, cosocleC22⋊#95C23
prime radical#16252C22
LCSG ▷ C22⋊#95 ▷ C25 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C22⋊#95 ◁ G
LECSG ▷ C22⋊#95 ▷ C25 ▷ C22 ▷ C2 ▷ C1

1050 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 24
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,6)(3,7)(4,8)(11,17,18,20)(14,22,23,24)(15,19);(1,2)(3,5)(4,7)(6,8)(9,10,13,16)(11,14)(12,19,21,15)(17,22)(18,23)(20,24);(1,3,5,2)(4,6,8,7)(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,22,19,24)

Polycyclic presentation:

2^10|a^2=i,b^2=ij,c^2=ghj,d^2=gj,e^2=gh,b^a=be,c^a=cd,c^b=cgj,d^a=dg,d^b=df,d^c=dgj,e^a=egh,e^b=egh,e^c=efi,e^d=eij,f^a=fi,f^c=fj,f^d=fj,g^b=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72238069

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