#72242271

order	12800
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ C₄×#1398

Subgroups

	subgroup	G/H
center	C₈	C₄×AGL(1,𝔽₅)²
commutator	C₄×C₅²	C₄²×C₈
Frattini	C₂×C₈	C₂×AGL(1,𝔽₅)²
Fitting	C₄×C₅²×C₈	C₄²
socle	C₂×C₅²	C₄×#346
prime radical	C₂²×C₅⋊D₅	C₂×M₃₂
cosocle	C₂×#11716583	C₂³
hypercenter	C₄×C₈	AGL(1,𝔽₅)²
hypocenter	C₅²	C₄×#1398
LCS	G ▷ C₄×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ C₄×C₈

1340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 160

Polycyclic presentation:

2^9,5^2|a^2=d,b^2=e,c^2=f,d^2=g,g^2=h,h^2=i,c^a=ch,d^c=di,f^a=fi,j^a=j^2,j^b=j^3,j^d=j^4,j^e=j^4,k^a=j^2k,k^b=k^3,k^d=jk,k^e=k^4

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242339
maximal normal subgroup of	#72242339

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