#72320178

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	#72240169
commutator, Frattini, cosocle	C₂×#535	C₂⁴
prime radical	#72318630	C₂
LCS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35527 ◁ G
LECS	G ▷ C₂×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4703 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:

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

Polycyclic presentation:

2^11|a^2=hjk,c^2=ghjk,i^2=k,b^a=bk,c^a=cg,c^b=chjk,d^a=dik,d^b=de,d^c=def,e^a=ek,e^c=eh,f^a=fhik,f^b=fhk,f^c=fi,g^d=ghi,g^e=gk,g^f=gjk,h^a=hk,h^c=hk,i^b=ik,i^c=ij,i^d=ik,i^f=ik,i^g=ik,j^a=jk,j^c=jk,j^d=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320243
maximal normal subgroup of	#72320243

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