Finite Group Databasebeta

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#72210089

order2048
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#346276
commutator, Frattini, cosocleC24×Q8C24
prime radicalC2×#60400C22
LCSG ▷ C24×Q8 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C24×Q8 ◁ G
LECSG ▷ C24×Q8 ▷ C23 ▷ C2 ▷ C1

8005 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 8005 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|a^2=f,b^2=e,c^2=eghk,d^2=eghijk,e^2=k,f^2=k,h^2=k,b^a=bg,c^a=cefgjk,c^b=cghk,d^a=dh,d^b=dfjk,d^c=deghj,e^a=ei,e^c=ei,e^d=ej,f^b=fj,f^c=fijk,f^d=fik,f^e=fk,g^a=gj,g^b=gi,h^a=hi,h^b=hk,h^c=hi,h^d=hj,h^f=hk,i^a=ik,j^b=jk

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72210092
maximal normal subgroup of#72210092
maximal quotient of#72266483, #72266500, #72266626, …

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