#72234841

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁶
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ C₂³×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

At least 21183 conjugacy classes of subgroups; only the 5162 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2)(3,4)(8,15)(10,18)(11,16)(19,20);(1,2)(3,4)(5,6)(7,10,14,18)(8,16)(9,12)(11,15)(13,19,17,20);(1,2)(3,4);(8,15)(11,16)(13,17)(19,20);(2,4)(7,13,14,17)(10,19,18,20);(5,7,9,14)(6,10,12,18)(8,17,15,13)(11,20,16,19)

Polycyclic presentation:

2^11|b^2=jk,e^2=jk,f^2=k,g^2=k,b^a=bjk,e^a=eh,e^b=eh,e^c=eh,e^d=ejk,f^a=fi,f^b=fik,f^e=fg,g^a=gj,g^b=gj,g^d=gk,g^f=gk,i^b=ik,i^e=ijk,i^g=ik,j^f=jk

Conjugacy classes & character table

194 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262638, #72263054, #72312045
maximal normal subgroup of	#72262638, #72263054
maximal quotient of	#72234842, #72260825
automorphism group of	#12765, #12766, #12968, …

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