Finite Group Databasebeta

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#72251485

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#116351
commutatorC23×D4C22×C4
Frattini, cosocleC24×D4C23
prime radicalC22×#535C22
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G
LECSG ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1

3471 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 3471 conjugacy classes of subgroups — too many to draw. Only its 155 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=df,d^2=j,b^a=bg,c^a=cd,c^b=cde,d^b=dh,d^c=dj,e^a=ehi,e^b=ehi,e^c=ehj,e^d=ej,f^c=fj,g^c=gi,g^d=gj,g^e=gj,h^a=hj,h^b=hj,h^c=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72251494, #72251580, #72251606, …
maximal normal subgroup of#72251494, #72251580, #72251606, …
maximal quotient of#72262390

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