#72235058

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#379
commutator	C₂⁵	C₂×C₄²
Frattini, cosocle	C₂⁷	C₂³
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁷ ▷ C₂³ ▷ C₁

10733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^10|a^2=d,b^2=i,c^2=deg,b^a=bf,c^a=ce,c^b=cf,d^b=dh,d^c=dhj,e^a=ehj,e^b=ei,e^c=ehj,f^a=fh,f^c=fhi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235061, #72236520, #72309387, …
maximal normal subgroup of	#72235061, #72236520, #72309387, …
maximal quotient of	#72236102, #72236151, #72236753

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