Finite Group Databasebeta

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#72249675

order10240
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC22#72209701
commutator, prime radical, cosocle, hypocenter#72241456C2
FittingH(5,𝔽4)D5
LCSG ▷ #72241456
UCSC1 ◁ C22

49971 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 49971 conjugacy classes of subgroups — too many to draw. Only its 13 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) = 128

Polycyclic presentation:

2,5,2^10|a^2=k,g^2=l,h^2=kl,i^2=kl,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=cefghk,g^b=ghij,g^c=gk,g^f=gl,h^a=cdefh,h^b=i,h^c=hl,h^d=hkl,h^e=hk,h^g=hkl,i^a=degjkl,i^b=gil,i^c=il,i^d=il,i^e=ikl,i^f=ik,i^g=il,i^h=il,j^a=cdghi,j^b=gijk,j^c=jl,j^f=jkl,j^h=jk,j^i=jkl

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72249859, #72257142

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