Finite Group Databasebeta

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#72256005

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×#534C22
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G
LECSG ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1

3651 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 3651 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)(3,8)(4,15,17,28)(5,20)(6,10)(7,11)(9,19,26,18)(12,25,29,32)(13,27)(14,24)(16,31,30,22)(21,23);(1,3)(2,8)(4,16,17,30)(5,13)(6,21)(7,14)(9,25,26,32)(10,27)(11,24)(12,18,29,19)(15,31,28,22)(20,23);(1,4,7,17)(2,9,11,26)(3,12,14,29)(5,18,6,19)(8,22,24,31)(10,28,20,15)(13,30,21,16)(23,32,27,25)

Polycyclic presentation:

2^10|a^2=e,b^2=e,c^2=j,f^2=j,g^2=j,b^a=bd,c^a=cdfj,c^b=cdgj,d^c=dh,e^c=ej,f^a=fj,f^b=fhi,f^c=fh,f^d=fj,g^a=gi,g^b=gj,g^c=gh,g^d=gj,h^a=hj,h^b=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72256008, #72256060, #72256091, …
maximal normal subgroup of#72256008, #72256060, #72256091, …
maximal quotient of#72265705

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