#72258228

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁶
socle	C₂²	C₂²×#2679
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

145519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 128

Polycyclic presentation:

2^11|e^2=k,f^2=k,h^2=k,c^b=ci,d^a=di,d^b=dik,e^a=eg,e^b=eghk,e^d=ek,f^a=fgh,f^b=fgj,f^c=fk,f^e=fk,g^b=gk,g^f=gk,h^a=hk,h^e=hk,j^a=jk,j^b=jk,j^e=jk,j^f=jk

Conjugacy classes & character table

194 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267247, #72267348
Fitting subgroup of	#72267247
maximal subgroup of	#72267247, #72267327
maximal normal subgroup of	#72267247, #72267327
maximal quotient of	#72259591

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