#72225506

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225505
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₄×M₆₄	C₂³
prime radical	#513082	C₂²
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ #149495 ◁ G
LECS	G ▷ C₄×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,4,14,6,16,30,17,9,18,7,36,8,13,28,11)(3,19,24,40,21,41,62,57,23,56,26,52,27,10,37,38)(5,31,29,55,34,47,63,45,33,43,32,15,35,53,22,54)(12,44,49,59,48,61,39,58,50,64,46,20,51,60,42,25);(2,10)(3,20)(4,28)(5,32)(6,8)(7,30)(11,38)(12,45)(13,19)(14,52)(15,51)(16,56)(17,57)(18,41)(21,58)(22,35)(23,59)(24,64)(25,27)(26,44)(29,33)(31,39)(34,63)(36,40)(37,60)(42,43)(46,53)(47,49)(48,55)(50,54)(61,62);(2,11,13,36,18,17,16,14)(3,21,23,27)(5,33)(10,38,19,40,41,57,56,52)(12,46,48,42,50,49,51,39)(15,31,45,53,55,43,54,47)(20,58,59,25)(22,63)(24,62,26,37)(29,32)(34,35)(44,60,64,61)

Polycyclic presentation:

2^11|a^2=f,c^2=e,d^2=ghi,e^2=hj,f^2=h,g^2=ij,h^2=k,i^2=jk,j^2=k,b^a=bd,c^a=cijk,d^a=dg,d^b=dghk,d^c=djk,e^a=ej,e^d=ek,f^b=fhk,g^a=gi,g^b=gi,g^c=gk,h^b=hk,i^a=ij,i^b=ij,j^a=jk,j^b=jk

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225510
maximal normal subgroup of	#72225510

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