Finite Group Databasebeta

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#72338670

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237744
commutator, Frattini, cosocle#1781C24
prime radical#72241342C2
LCSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #20585 ◁ G
LECSG ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

2697 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 2697 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|c^2=eh,e^2=k,f^2=h,b^a=bgj,c^a=cf,c^b=ce,d^a=dh,d^b=dijk,d^c=dijk,e^a=eg,e^b=ek,e^c=ejk,e^d=ek,f^a=fh,f^b=fgk,f^c=fghjk,f^e=fij,g^c=gi,g^d=gk,g^f=gjk,h^b=hjk,h^c=hjk,h^e=hk,i^a=ij,i^f=ik,j^c=jk

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338677, #72338741, #72338805
maximal normal subgroup of#72338677, #72338741, #72338805

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