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#72268128

order10240
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22#72209701
commutator, prime radical, cosocle, hypocenter#72268127C2
Fitting#72268062D5
LCSG ▷ #72268127
UCSC1 ◁ C22

At least 7111 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has at least 7111 conjugacy classes of subgroups — too many to draw. Only its 14 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:

2,5,2^10|a^2=kl,g^2=k,h^2=k,j^2=k,b^a=b^4,c^a=cf,c^b=d,d^a=cef,d^b=cde,e^a=cd,e^b=f,f^b=cd,g^a=ghi,g^b=gh,g^c=gl,g^d=gl,g^e=gl,h^a=jk,h^b=ij,h^f=hl,h^g=hk,i^a=hijk,i^b=gik,i^d=il,i^e=il,i^f=il,i^g=ik,i^h=ik,j^a=hk,j^b=gijk,j^c=jl,j^d=jl,j^f=jl,j^h=jk

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268145

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", "l");;
gap> coll := SingleCollector(F, [2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.11*F.12);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2^4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 6, 2, F.3*F.4);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.12);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 2, F.7*F.9*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 9, 6, F.9*F.12);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.8*F.11);;
gap> SetConjugate(coll, 10, 2, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.12);;
gap> SetConjugate(coll, 10, 4, F.10*F.12);;
gap> SetConjugate(coll, 10, 6, F.10*F.12);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);