Finite Group Databasebeta

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#72310520

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72266164
commutator#2308C22×C4
Frattini, cosocle#35346C23
prime radical#6432953C22
LCSG ▷ #2308 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #35346 ◁ G
LECSG ▷ #35346 ▷ C25 ▷ C23 ▷ C2 ▷ C1

3132 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 3132 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,8,13,23)(4,7)(5,9)(6,10)(11,24,27,22)(12,18,14,29)(15,19)(16,20)(17,25)(21,28,31,26)(30,32);(2,7)(3,11)(8,21)(9,19,25,32)(10,20)(12,28,14,26)(13,27)(15,30)(18,22,29,24)(23,31);(1,3)(2,8,10,23)(4,14)(5,11)(6,13)(7,18,20,29)(9,24,25,22)(12,16)(15,26)(17,27)(19,31,32,21)(28,30)

Polycyclic presentation:

2^11|a^2=i,b^2=e,c^2=j,b^a=bd,c^a=chij,c^b=cef,d^b=dg,d^c=dgjk,e^a=eg,e^c=eh,e^d=ei,f^a=fi,f^b=fhj,f^c=fhj,f^e=fj,g^b=gi,g^f=gk,h^b=hj,h^d=hk,i^c=ik,j^a=jk

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72310522, #72337732, #72337733
maximal normal subgroup of#72310522, #72337732, #72337733

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