#72283746

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	#72234509
commutator, Frattini, cosocle	#2309	C₂⁴
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35626 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

5887 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)(3,13)(4,18)(5,23)(6,12)(7,8)(9,16)(10,22)(11,27)(14,42)(15,43)(17,39)(19,54)(20,52)(21,53)(24,32)(25,37)(26,41)(28,33)(29,44)(30,38)(31,40)(34,49)(35,48)(36,46)(45,61)(47,62)(50,51)(55,60)(56,59)(57,63)(58,64);(2,8)(3,14)(6,28)(9,34)(10,18)(11,23)(13,42)(15,47)(16,49)(17,29)(19,45)(24,46)(26,31)(32,36)(35,57)(37,52)(38,53)(39,44)(43,62)(48,63)(50,64)(51,58)(54,61)(59,60);(1,3)(2,9)(4,19)(5,24)(6,29)(7,17)(8,32)(10,35)(11,39)(12,34)(13,23)(14,28)(15,22)(16,27)(18,51)(20,50)(21,45)(25,48)(26,49)(30,47)(31,46)(33,36)(37,43)(38,58)(40,44)(41,42)(52,54)(53,57)(55,64)(56,63)(59,61)(60,62);(1,4)(2,10,8,18)(3,15)(5,25,27,20)(6,30,28,21)(7,22)(9,35,32,51)(11,37)(12,38)(13,43)(14,45,29,47)(16,50,24,48)(17,19)(23,52)(26,56)(31,55)(33,53)(34,57)(36,58)(39,54)(40,60,41,59)(42,61,44,62)(46,63)(49,64)

Polycyclic presentation:

2^11|d^2=hi,e^2=h,f^2=h,b^a=bj,c^a=ce,c^b=cf,d^a=dg,d^b=dghk,d^c=dhik,e^a=eh,e^b=ejk,e^c=eh,e^d=ei,f^a=fjk,f^b=fh,f^c=fh,f^d=fi,f^e=fk,g^c=gik,g^d=gj,g^e=gjk,g^f=gj,h^d=hk,i^a=ij,i^b=ijk,i^d=ik,i^e=ik,i^f=ik,j^c=jk

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285103
maximal normal subgroup of	#72285103

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