Finite Group Databasebeta

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#72310538

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72263077
commutator#2308C22×C4
Frattini, cosocle#35346C23
prime radical#72237724C2
LCSG ▷ #2308 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #35346 ◁ G
LECSG ▷ #35346 ▷ C25 ▷ C23 ▷ C2 ▷ C1

4723 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 4723 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)(4,7)(5,17)(6,10)(9,19)(11,26)(12,24)(13,25)(14,20)(15,27)(16,22)(18,21)(23,29)(28,30)(31,32);(2,7)(3,11)(8,23,25,32)(9,27,17,21)(10,22)(12,29,14,31)(13,30)(15,18)(20,28)(24,26);(1,3)(2,8,10,25)(4,14,16,12)(5,11)(6,13)(7,20)(9,28,17,26)(15,31,18,29)(19,30)(21,23)(22,24)(27,32)

Polycyclic presentation:

2^11|b^2=e,c^2=h,b^a=bd,c^a=cik,c^b=cef,d^b=dg,d^c=dg,e^a=eg,e^c=eh,e^d=ei,f^a=fik,f^b=fhj,f^c=fh,f^d=fk,f^e=fj,g^b=gi,g^c=gk,g^f=gk,h^a=hk,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#72310541, #72337746, #72337775
maximal normal subgroup of#72310541, #72337746, #72337775

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