#72309253

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#152049
commutator	C₄×M₁₆	C₂×C₄²
Frattini, cosocle	C₂×#3151	C₂³
prime radical	C₂×#19745	C₂²
LCS	G ▷ C₄×M₁₆ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₄³ ◁ #6376438 ◁ G
LECS	G ▷ C₂×#3151 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

3315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

2^11|a^2=d,b^2=fg,c^2=dhi,d^2=j,f^2=ij,h^2=jk,i^2=k,b^a=bf,c^a=cj,c^b=ce,d^b=dijk,e^a=ej,e^b=ej,f^b=fh,f^c=fj,f^e=fk,g^a=ghij,g^b=gh,g^d=gj,g^f=gij,h^b=hi,h^g=hjk,i^b=ij,i^g=ik,j^b=jk

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309267
maximal normal subgroup of	#72309267

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