#72225019

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72225017
commutator, Frattini, cosocle	C₂²×C₁₆	C₂⁵
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #60204 ◁ #7622661 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

6734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=g,e^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bgk,d^a=df,d^b=dh,d^c=dh,e^a=ejk,f^a=fk,f^c=fk,g^b=gi,g^d=gk,g^e=gk,h^a=hk,i^b=ij,j^b=jk

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225136, #72343072
maximal normal subgroup of	#72225136, #72343072

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