#72247472

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂⁴×#120
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

Only the 38403 normal subgroups are shown; non-normal subgroups are omitted.

Full subgroup data →

Presentation

Polycyclic presentation:

2^10|b^2=j,b^a=bj,d^c=dj,e^a=eh,e^b=ei,e^d=ej,g^a=ghj,g^b=gi,g^f=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266787, #72266807, #72266815, …
maximal normal subgroup of	#72266787, #72266807, #72266815, …
maximal quotient of	#72252431, #72267098

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