Finite Group Databasebeta

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#72285566

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237777
commutator#1781C22×C4
Frattini, cosocle#20585C23
prime radical#72241347C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2541 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 2541 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,14)(4,10,5,11)(6,7)(8,13)(9,16)(12,26)(15,28,31,18)(17,32,25,27)(19,30)(20,29)(21,22,23,24);(1,3,7,18)(2,9,12,17,13,29,6,25)(4,20,22,27)(5,16,24,32)(8,19,26,28)(10,14,23,15,11,30,21,31);(1,4)(2,10,13,11)(3,15)(5,8)(6,21)(7,24,26,22)(9,27,29,32)(12,23)(14,28)(16,25,20,17)(18,30)(19,31)

Polycyclic presentation:

2^11|a^2=gik,b^2=f,c^2=ghik,d^2=hijk,e^2=h,f^2=hi,b^a=bd,c^a=cghjk,c^b=ce,d^a=dh,d^b=dhj,d^c=dg,e^a=egj,e^b=egjk,e^c=ehik,e^d=eij,f^a=fi,f^c=fghjk,f^d=fj,f^e=fik,g^a=gj,g^b=gi,g^c=gj,g^d=gj,g^e=gjk,g^f=gjk,h^a=hjk,h^b=hjk,h^c=hk,h^d=hk,h^f=hk,i^a=ij,i^b=ijk,i^c=ijk,i^d=ik,i^e=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338677, #72338761, #72338824
maximal normal subgroup of#72338677, #72338761, #72338824

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