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#72237358

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72236911
commutator, Frattini, cosocleC2×#2849C23
prime radical#72237355C2
LCSG ▷ C2×#2849 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2849 ◁ G
LECSG ▷ C2×#2849 ▷ C25 ▷ C23 ▷ C2 ▷ C1

3116 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 3116 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,6)(4,14)(5,9)(7,11)(8,17)(10,22)(12,18)(13,19)(15,30)(16,29)(20,26)(21,25)(23,28)(24,31)(27,32);(1,3,6,2)(4,15,17,23,18,29,31,32)(5,13,19,9)(7,10,22,11)(8,16,14,28,24,30,12,27)(20,25,21,26);(1,4,7,18)(2,8)(3,12)(5,16,20,30)(6,17,22,31)(9,23)(10,14)(11,24)(13,15)(19,28,21,27)(25,29)(26,32)

Polycyclic presentation:

2^11|b^2=f,c^2=jk,d^2=k,e^2=k,f^2=ij,b^a=bf,c^a=ce,c^b=cd,d^a=dg,d^b=dgh,e^a=ek,e^b=ehijk,e^d=ejk,f^a=fij,f^c=fghk,f^d=fjk,f^e=fijk,g^b=gi,g^c=gj,g^e=gk,g^f=gk,h^a=hijk,h^b=hijk,h^c=hj,i^a=ik,i^b=ik,i^c=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237404
maximal normal subgroup of#72237404

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.10*F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.10*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);