Finite Group Databasebeta

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#72267809

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC23#2021
commutator, Frattini, cosocleC23×C4⋊C4C23
prime radical#35223C22
LCSG ▷ C23×C4⋊C4 ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C23×C4⋊C4 ▷ C24 ▷ C2 ▷ C1

2241 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 2241 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=hij,b^2=i,c^2=eij,d^2=gj,f^2=j,b^a=bd,c^a=ce,c^b=cefj,d^a=dg,d^b=dgj,d^c=dh,e^b=ei,f^a=fhij,f^b=fij,f^c=fij,f^d=fj,h^b=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267813, #72267855
maximal normal subgroup of#72267813, #72267855
maximal quotient of#72267821, #72267861

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