Finite Group Databasebeta

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#72337728

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72266698
commutatorC22×#120C22×C4
Frattini, cosocle#35346C23
prime radical#72337171C2
LCSG ▷ C22×#120 ▷ C25 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C25 ◁ #35346 ◁ G
LECSG ▷ #35346 ▷ C25 ▷ C23 ▷ C2 ▷ C1

5003 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 5003 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=fhijk,b^a=bdf,c^a=ce,c^b=ceij,d^b=djk,d^c=dg,e^a=eg,e^b=eh,e^d=ei,f^a=fj,f^b=fjk,f^c=fhk,f^d=fk,f^e=fj,g^a=gi,g^b=gi,g^f=gk,h^a=hj,h^b=hj,h^d=hk,i^b=ik,j^a=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72337729, #72337762, #72337775
maximal normal subgroup of#72337729, #72337762, #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, 1, F.4);;
gap> SetPower(coll, 2, F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);