#72319277

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 ◁ C₂×#17320 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5541 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)(5,6,9,12,10,13,14,7)(8,11,17,19,16,18,20,15)(22,26,29,31)(23,30)(24,33)(27,32)(28,34,36,35);(5,6,9,12,10,13,14,7)(8,11,17,19,16,18,20,15)(21,22,27,35,33,36,32,26,25,29,30,34,24,28,23,31);(5,7,14,13,10,12,9,6)(8,15,20,18,16,19,17,11)(22,28,29,36)(23,32)(26,34,31,35)(27,30);(5,8)(6,11)(7,15)(9,17)(10,16)(12,19)(13,18)(14,20);(2,4)(8,16)(11,18)(15,19)(17,20)(22,29)(26,31)(28,36)(34,35)

Polycyclic presentation:

2^11|a^2=f,b^2=fgjk,c^2=fi,f^2=i,g^2=j,j^2=k,b^a=bg,c^b=cjk,e^a=eh,e^b=ek,e^d=ei,f^b=fk,g^a=gj,g^b=gk,g^c=gk,j^a=jk

Conjugacy classes & character table

182 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, 1, F.6);;
gap> SetPower(coll, 2, F.6*F.7*F.10*F.11);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);