#72280959

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂⁵×D₄
commutator	C₄	C₂⁶×C₄
Frattini, cosocle	C₂×C₄	C₂⁷
prime radical	C₂×#63091	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#585 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Only the 60197 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|c^2=j,d^2=h,f^2=ij,i^2=j,c^a=cj,d^b=dj,e^a=ej,e^c=ej,f^b=fj,f^c=fi,f^d=fij,g^b=gj,i^c=ij,i^d=ij

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281386, #72281426, #72282174, …
maximal normal subgroup of	#72281386, #72281426, #72282174, …
maximal quotient of	H(5,𝔽₂)×#442, #72343813

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