#72225173

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#140526
commutator	C₄×C₁₆	C₂²×C₈
Frattini, cosocle	C₂×#1311	C₂³
prime radical	#491485	C₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ C₈×#346 ◁ G
LECS	G ▷ C₂×#1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

843 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:

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

Polycyclic presentation:

2^11|a^2=d,c^2=f,d^2=g,e^2=h,f^2=j,g^2=j,h^2=i,i^2=k,b^a=be,c^a=cf,c^b=cfj,d^b=dik,e^a=eh,e^b=ehik,e^d=ek,f^a=fj,f^b=fj,g^b=gk,h^a=hi,h^b=hik,i^a=ik,i^b=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225181, #72225249, #72225255
maximal normal subgroup of	#72225181, #72225249, #72225255

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