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#72227942

order2048
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#245225
commutator, Frattini, cosocleC22×#127C24
LCSG ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G
LECSG ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1

11045 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 11045 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|b^a=bk,c^a=cg,c^b=ce,d^b=df,d^c=dijk,e^a=eh,e^d=ei,f^c=fi,f^e=fjk,g^b=ghk,g^d=gj,g^e=gk,g^f=gj,h^c=hk,h^d=hj,i^a=ij,i^b=ijk

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227953

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