Finite Group Databasebeta

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#72286251

order2048
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C42

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC2#72211186
commutator, Frattini, cosocleC22×H(5,𝔽2)C24
prime radical#72213228C2
LCSG ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G
LECSG ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1

8936 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 8936 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 32

Polycyclic presentation:

2^11|a^2=jk,b^2=gjk,c^2=h,d^2=h,e^2=k,b^a=bg,c^a=ce,c^b=cf,d^a=dgijk,d^b=dghi,d^c=dhk,e^b=ek,e^c=ej,e^d=ei,f^a=fij,f^b=fijk,f^c=fjk,f^d=fik,g^c=gij,g^d=gj,g^f=gk,h^a=hjk,h^b=hjk,h^e=hk,h^f=hk,i^b=ik,i^c=ik,i^d=ik,j^c=jk

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72338974, #72338982, #72338990, …
maximal normal subgroup of#72338974, #72338982, #72338990, …

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