#72267830

order	2560
automorphism	?
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#160295 ⋊ C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#17102⋊C₅
commutator, Fitting, cosocle, hypocenter	#160295	C₅
Frattini, socle	C₂⁵	C₂⁴⋊C₅
LCS	G ▷ #160295
UCS	C₁ ◁ C₂

581 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 40

Polycyclic presentation:

5,2^9|b^2=f,c^2=fj,d^2=h,e^2=fgij,b^a=dfh,c^a=bcefj,c^b=cfg,d^a=cdfgh,d^b=dfgi,d^c=dfgh,e^a=def,e^b=ef,e^c=efj,e^d=eh,f^a=h,g^a=hi,h^a=gj,i^a=fghi,j^a=hij

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267831, #72267871
maximal subgroup of	#72267831, #72267871
maximal normal subgroup of	#72267831, #72267871
maximal quotient of	#72267832

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