#72225178

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#9837
commutator	C₈²	C₂²×C₈
Frattini, cosocle	C₄×C₈²	C₂³
socle	C₂²	#102117
prime radical	#491465	C₄
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,3,7,5,9,11,10)(4,13,12,8,15,6,14,16)(17,18,21,28,24,30,23,42,26,31,22,34,25,19,32,29)(20,35,37,33,39,48,44,27,41,36,43,46,40,45,38,47);(2,6)(3,11)(4,14)(7,8)(9,13)(10,16)(12,15)(17,19)(18,24)(20,36)(21,42)(22,29)(23,34)(25,31)(26,30)(27,44)(28,32)(33,37)(35,41)(38,47)(39,45)(40,48)(43,46);(17,20,23,38,25,40,21,43,26,41,32,44,24,39,22,37)(18,27,28,35,30,46,42,48,31,47,34,36,19,33,29,45)

Polycyclic presentation:

2^11|a^2=d,c^2=f,d^2=g,e^2=h,f^2=i,g^2=k,h^2=j,i^2=k,b^a=be,c^a=cf,c^b=cfik,d^b=dj,d^c=dk,e^a=eh,e^b=ehj,f^a=fi,f^b=fik,h^a=hj,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225190, #72225236
maximal normal subgroup of	#72225190, #72225236
maximal quotient of	#72225181

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