Finite Group Databasebeta

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#72338774

order2048
nilpotency class6
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72237593
commutator#2308C22×C4
Frattini, cosocle#20585C23
prime radical#72240637C2
LCSG ▷ #2308 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #6432953 ◁ G
LECSG ▷ #20585 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1

4116 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 4116 conjugacy classes of subgroups — too many to draw. Only its 90 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)(5,9)(6,10)(13,28)(15,29)(16,22)(17,25)(20,24)(23,30);(1,3,4,12,6,14,16,26)(2,7,8,19,10,21,22,31)(5,13,15,27,17,28,30,11)(9,24,23,18,25,20,29,32);(3,11)(7,18)(8,22)(9,25)(12,28)(13,26)(14,27)(15,30)(19,20)(21,32)(23,29)(24,31)

Polycyclic presentation:

2^11|b^2=d,d^2=k,e^2=hk,b^a=bf,c^a=chijk,c^b=cdehk,d^a=dik,d^c=dg,e^a=eh,e^b=eghi,e^c=eghijk,e^d=eij,f^b=fik,f^c=fhijk,g^a=gjk,g^b=gi,g^e=gjk,g^f=gk,h^b=hj,h^c=hj,h^d=hk,i^c=ij,i^e=ik,j^b=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338791, #72338807
maximal normal subgroup of#72338791, #72338807

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