#72209978

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

solvable

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄²	(C₂⁴⋊C₅)⋊C₂
commutator, hypocenter	#564⋊C₅	C₂²×C₄
Frattini, socle	C₂³	C₂²×(C₂⁴⋊C₅)⋊C₂
Fitting	C₂×#62694	D₅
prime radical	C₂×#62694⋊C₅	C₂
cosocle	C₂×#564⋊C₅	C₂³
LCS	G ▷ #564⋊C₅
UCS	C₁ ◁ C₂×C₄²

6307 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 6307 conjugacy classes of subgroups — too many to draw. Only its 97 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:

2^4,5,2^6|a^2=j,b^2=jk,c^2=d,f^2=k,g^2=k,h^2=j,i^2=j,e^c=e^4,f^c=gk,f^e=g,g^c=fk,g^e=fgh,g^f=gj,h^c=gik,h^e=i,h^f=hk,i^c=fhk,i^e=fg,i^f=ij,i^g=ik,i^h=ijk

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72209980

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