#72225194

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#123285
commutator	C₈×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₈×C₁₆	C₂³
prime radical	C₂×#508	D₈
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6)(7,11)(8,12)(9,14)(10,16)(13,32)(15,22)(17,23)(18,25)(19,31)(20,26)(21,28)(24,30)(27,35)(29,34)(33,36);(1,2,4,3)(7,18)(8,22)(9,23)(11,25)(12,15)(13,28)(14,17)(19,24)(20,36)(21,32)(26,33)(27,29)(30,31)(34,35);(1,3,4,2)(6,11,16,28)(7,18)(8,22)(9,23)(12,27,30,33)(13,15,25,31)(14,29,17,26)(19,24)(20,36)(21,32)(34,35)

Polycyclic presentation:

2^11|b^2=f,c^2=fi,d^2=gi,e^2=hjk,g^2=i,h^2=jk,j^2=k,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dg,e^a=ehk,e^b=eh,e^c=eh,g^a=gj,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hj,i^a=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225219, #72226838, #72226963
maximal normal subgroup of	#72225219, #72226838, #72226963
maximal quotient of	#72227135

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