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#72268804

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227479
commutator, Frattini, cosocleC2×#2614C23
prime radical#4930596C22
LCSG ▷ C2×#2614 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#2614 ◁ G
LECSG ▷ C2×#2614 ▷ C25 ▷ C23 ▷ C2 ▷ C1

4832 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 4832 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,12)(3,13,17,40)(4,18,23,38)(5,24,28,10)(6,29,33,11)(8,30,36,16)(9,26,22,20)(14,44,47,64)(15,48,25,43)(19,54,34,63)(21,59,31,53)(27,39,32,41)(35,52,50,49)(37,58,61,60)(42,62,56,45)(46,55,51,57);(1,3,2,8,7,17,12,36)(4,19,22,47,23,34,9,14)(5,25,10,35,28,15,24,50)(6,30,11,40,33,16,29,13)(18,45,20,57,38,62,26,55)(21,56,37,46,31,42,61,51)(27,49,39,43,32,52,41,48)(44,59,63,58,64,53,54,60);(1,4,12,22,7,23,2,9)(3,14,8,34,17,47,36,19)(5,26,24,18,28,20,10,38)(6,31,29,61,33,21,11,37)(13,42,30,46,40,56,16,51)(15,45,50,55,25,62,35,57)(27,58,41,53,32,60,39,59)(43,44,52,54,48,64,49,63)

Polycyclic presentation:

2^11|a^2=k,b^2=f,c^2=fj,d^2=k,e^2=k,f^2=k,b^a=be,c^a=cd,c^b=cfij,d^a=dk,d^b=dg,d^c=dh,e^a=ek,e^b=eh,e^c=eghj,e^d=ek,f^a=fhk,f^c=fk,f^d=fi,f^e=fijk,g^a=gk,g^b=gi,g^c=gj,g^f=gk,h^b=hijk,h^c=hi,i^b=ik,j^b=jk,j^c=jk

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72268809
maximal subgroup of#72268809
maximal normal subgroup of#72268809
maximal quotient of#72268874

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