Finite Group Databasebeta

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#72338874

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72227478
commutator#2309C42
Frattini, prime radical, cosocle#6628901C22
LCSG ▷ #2309 ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6628901 ◁ G
LECSG ▷ #6628901 ▷ C23×D4 ▷ C24 ▷ C22 ▷ C2 ▷ C1

4729 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 4729 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=e,b^2=d,c^2=hik,d^2=hk,g^2=k,b^a=bc,c^a=cg,c^b=cf,d^a=dfhij,d^c=di,e^b=eghi,e^c=ehik,e^d=ehk,f^a=fh,f^b=fi,f^c=fjk,f^d=fjk,f^e=fj,g^a=ghi,g^b=ghjk,g^d=gk,g^e=gk,g^f=gk,h^a=hj,h^c=hk,i^a=ijk,i^b=ijk,i^c=ik,i^d=ik,j^b=jk

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338875, #72338890, #72338899, …
maximal normal subgroup of#72338875, #72338890, #72338899, …

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