#72242602

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #21233

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₅⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₅²×D₈	C₂×C₄
socle	C₂²×C₅²	C₂×C₄×D₄
prime radical	C₂×D₅²×D₈	C₂
cosocle	C₂×C₄×C₅⋊D₅	C₂⁴
hypercenter	C₂×D₈	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#21233
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

3956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 320

Polycyclic presentation:

2^8,5^2|a^2=e,b^2=ef,c^2=gh,d^2=gh,g^2=h,b^a=bf,c^a=cg,d^a=dgh,d^b=dh,g^a=gh,i^a=i^3,i^b=i^3,i^c=i^4,i^e=i^4,j^a=j^3,j^b=j^3,j^c=i^4j,j^e=j^4

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72269238
maximal normal subgroup of	#72269238

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