#72319181

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#442
commutator	C₂²×C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁵
prime radical	#72319173	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ #181905 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5355 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^11|b^2=f,c^2=fijk,f^2=i,g^2=jk,j^2=k,b^a=bg,c^a=cj,c^b=cgj,e^a=eh,e^b=ei,e^c=eik,e^d=ei,f^a=fk,f^c=fk,g^a=gj,g^b=gjk,g^c=gjk,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319695
maximal normal subgroup of	#72319695

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