#72226896

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72226895
commutator	C₄×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
prime radical	#72225569	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ #493217 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=g,c^2=e,d^2=fh,f^2=hj,g^2=ik,h^2=jk,i^2=k,j^2=k,b^a=bghj,c^a=cj,c^b=cijk,d^a=df,d^b=df,d^c=djk,e^a=ek,e^d=ek,f^a=fh,f^b=fh,f^c=fk,g^b=gi,h^a=hj,h^b=hj,i^b=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226910, #72227175, #72227179
maximal normal subgroup of	#72226910, #72227175, #72227179

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