#72225255

order	4096
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₁₆

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72216172
commutator	C₈×C₁₆	C₂²×C₈
Frattini, cosocle	C₈×#346	C₂³
prime radical	#72216261	C₄
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₈² ◁ #72216161 ◁ G
LECS	G ▷ C₈×#346 ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1330 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:

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

Polycyclic presentation:

2^12|a^2=d,c^2=eh,d^2=g,e^2=hj,f^2=i,h^2=jl,i^2=k,j^2=l,b^a=bf,c^a=chj,c^b=ce,d^b=di,d^c=dj,e^a=ejl,e^b=eh,e^d=el,f^a=fk,f^b=fik,g^b=gk,g^c=gl,h^a=hl,h^b=hj,i^b=ik,j^b=jl

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225268
maximal normal subgroup of	#72225268

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5*F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 8, F.10*F.12);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);