#72242672

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄×C₅

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₂×#405

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#1044
commutator	C₂×C₅²	C₂²×C₄²
Frattini	C₂²	C₂³×#1044
Fitting	C₅²×#117	C₄
socle	C₂²×C₅²	C₂³×C₄
prime radical	C₂³×C₅⋊D₅	C₂×C₄
cosocle	C₂×#1039	C₂⁴
hypercenter	#117	#1044
hypocenter	C₅²	C₂×#405
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ #117

1514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 160

Polycyclic presentation:

2^7,5^2|a^2=f,b^2=e,d^2=g,e^2=g,b^a=bg,c^a=cg,h^b=h^2,h^e=h^4,i^b=h^3i^3,i^e=i^4

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242675, #72242688, #72242691
maximal normal subgroup of	#72242675, #72242688, #72242691

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