#72285916

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

5805 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)(4,9)(5,10,25,37)(6,26)(7,12)(8,16)(11,28)(14,31)(15,46,22,41)(17,29)(18,42)(19,33,23,55)(20,38)(21,35)(24,39,56,36)(27,34)(30,47,61,49)(32,50)(40,45)(43,62,53,60)(44,58)(48,59,54,57)(51,64,52,63);(1,3,4,16)(2,8,9,13)(5,22,23,49,25,15,19,47)(6,17,27,45)(7,18,21,50)(10,30,33,41,37,61,55,46)(11,31,38,58)(12,32,35,42)(14,28,44,20)(24,48,51,53,56,54,52,43)(26,40,34,29)(36,62,64,57,39,60,63,59);(1,4,7,21)(2,9,12,35)(3,14)(5,19)(6,27,28,20)(8,29)(10,33)(11,38,26,34)(13,40)(15,43,22,53)(16,44)(17,50)(18,45)(23,25)(24,51)(30,59,61,57)(31,42)(32,58)(36,63)(37,55)(39,64)(41,62,46,60)(47,48,49,54)(52,56);(1,5)(2,10,12,37)(3,15)(4,19,21,23)(6,24,28,56)(7,25)(8,30,32,61)(9,33)(11,39)(13,41)(14,43)(16,49,50,47)(17,48,44,54)(18,22)(20,52)(26,36)(27,51)(29,57,58,59)(31,62)(34,63,38,64)(35,55)(40,60)(42,46)(45,53)

Polycyclic presentation:

2^11|a^2=h,b^2=e,c^2=hj,d^2=hik,e^2=h,f^2=h,b^a=be,c^a=cjk,c^b=cf,d^a=dg,d^b=dghij,d^c=dghij,e^a=eh,e^c=ej,e^d=ei,f^a=fj,f^b=fhj,f^c=fhk,f^d=fik,f^e=fk,g^a=gk,g^b=gijk,g^c=gjk,g^d=gj,g^e=gjk,g^f=gj,h^d=hk,i^a=ij,i^b=ij,i^c=ij,i^d=ik,i^e=ik,i^f=ik,j^b=jk

Conjugacy classes & character table

68 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.8);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.8*F.9*F.11);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.10*F.11);;
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.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*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);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
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, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);