#72258176

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#1776
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×C₈²	C₂³
socle, prime radical	C₂³	#8286
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #126522 ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 2048

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,6,5,7,8,3)(10,14,16,22)(11,18)(15,20)(17,24,19,23)(26,32,27,35)(28,34)(29,37)(30,33)(36,39,38,40);(1,2,4,6,5,7,8,3)(10,14,16,22)(11,18)(15,20)(17,24,19,23)(25,26,33,36,31,27,30,38)(28,39,37,32,29,40,34,35);(1,3,8,7,5,6,4,2)(9,10,15,23,21,22,18,17,13,16,20,24,12,14,11,19)(25,27,37,40,33,36,29,32,31,26,34,39,30,38,28,35)

Polycyclic presentation:

2^11|a^2=d,b^2=dij,c^2=dehj,d^2=g,e^2=h,f^2=ijk,h^2=j,i^2=k,b^a=bf,c^a=cfhij,c^b=cfh,d^b=dk,d^c=dj,e^a=ei,e^b=ei,e^c=ej,f^a=fijk,f^b=fijk,f^c=fk,h^a=hk,h^b=hk,i^a=ik,i^b=ik

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258190
maximal normal subgroup of	#72258190
maximal quotient of	#72258177

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