Finite Group Databasebeta

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#72226628

order2560
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#17102 ⋊ D5
derived#17102⋊C5 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#17102⋊C5C2
Frattini, socleC24(C24⋊C5)⋊C2
Fitting#17102D5
LCSG ▷ #17102⋊C5
UCSC1

501 conjugacy classes of subgroups; only the 5 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 5 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) = 20
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 20 points:

(1,5,4,3)(2,12)(6,8,11,9)(7,13,14,18)(10,17)(15,19,16,20);(1,2,16,13,6)(3,8,14,19,17)(4,10,7,20,11)(5,9,15,18,12)

Polycyclic presentation:

2,5,2^8|a^2=hij,e^2=g,f^2=gi,b^a=b^4ij,c^a=dfg,c^b=ce,d^a=efhj,d^b=efgi,d^c=di,e^a=cdefij,e^b=cdefgh,e^c=eg,e^d=eh,f^a=cefgj,f^b=defgi,f^c=fgij,f^d=fgi,f^e=fi,g^a=ghij,g^b=ghij,h^a=ghj,h^b=ghj,i^a=gh,i^b=j,j^a=hi,j^b=gj

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72237378, #72237379
maximal normal subgroup of#72237378, #72237379
maximal quotient of#72226629, #72226655, #72238094, …

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