#72227153

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72227152
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	#512952	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ #479908 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1225 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:

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

Polycyclic presentation:

2^11|a^2=g,b^2=g,d^2=e,e^2=h,f^2=ij,g^2=k,i^2=jk,j^2=k,b^a=bf,c^a=cgjk,c^b=cfgij,d^a=dk,d^b=dij,e^b=ejk,f^a=fi,f^b=fi,f^c=fi,f^d=fj,f^e=fk,g^c=gk,h^b=hk,i^a=ij,i^b=ij,i^c=ij,i^d=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

137 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227158, #72321568
maximal normal subgroup of	#72227158, #72321568

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