#72241399

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72216427
commutator	C₂²×H(5,𝔽₂)	C₂²×C₄
Frattini, cosocle	C₂×H(7,𝔽₂)	C₂³
prime radical	#72213228	C₂
LCS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁵ ◁ C₂×H(7,𝔽₂) ◁ G
LECS	G ▷ C₂×H(7,𝔽₂) ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁

8413 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 8413 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 32

Polycyclic presentation:

2^11|a^2=d,b^2=j,c^2=j,e^2=k,f^2=k,b^a=be,c^a=cf,c^b=cj,d^b=dh,d^c=dg,e^a=ehk,e^c=ei,e^d=ek,f^a=fgk,f^b=fi,f^d=fk,f^e=fk,g^a=gk,g^b=gj,g^e=gk,h^a=hk,h^c=hj,h^f=hk,i^a=ij,i^d=ik,j^a=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72241400, #72252411, #72267055
maximal subgroup of	#72241400, #72252402, #72252409, …
maximal normal subgroup of	#72241400, #72252402, #72252409, …

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