#72319822

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72318195
commutator, Frattini, cosocle	C₂²×#127	C₂⁴
prime radical	C₂²×#453	C₂³
LCS	G ▷ C₂²×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#117 ◁ #34210 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4243 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)(3,8)(4,9)(5,21,24,10)(6,27)(7,11)(12,28)(13,29)(14,43,44,36)(15,31)(16,46)(17,32)(18,33,23,37)(19,34)(20,52)(22,30,45,58)(25,39,26,38)(35,51)(40,62,63,59)(41,56)(42,57)(47,55)(48,54,53,64)(49,60,50,61);(1,3)(2,8)(4,13)(5,22,25,14)(6,17,28,16)(7,15)(9,29)(10,36,38,58)(11,31)(12,32,27,46)(18,48,49,63)(19,42)(20,41,51,47)(21,43,39,30)(23,53,50,40)(24,45,26,44)(33,59,60,54)(34,57)(35,56,52,55)(37,62,61,64);(1,4,7,19)(2,9,11,34)(3,13,15,42)(5,23)(6,20,28,51)(8,29,31,57)(10,37)(12,35,27,52)(14,40)(16,47,17,41)(18,24)(21,33)(22,53)(25,50)(26,49)(30,54)(32,56,46,55)(36,62)(38,61)(39,60)(43,59)(44,63)(45,48)(58,64);(1,5,7,24)(2,10,11,21)(3,14,15,44)(4,18,19,23)(6,25,28,26)(8,30,31,58)(9,33,34,37)(12,38,27,39)(13,40,42,63)(16,45,17,22)(20,49,51,50)(29,54,57,64)(32,43,46,36)(35,60,52,61)(41,53,47,48)(55,62,56,59)

Polycyclic presentation:

2^11|a^2=f,b^2=h,c^2=fk,d^2=k,i^2=k,b^a=bh,c^a=cf,d^a=dfg,d^b=dgjk,d^c=de,e^b=ek,e^c=ek,f^d=fk,g^a=gk,g^b=gi,g^c=gk,g^d=gk,h^d=hi,h^g=hjk,i^a=ij,i^b=ij,i^d=ik,i^g=ik,i^h=ik,j^b=jk,j^d=jk

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

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