#72287261

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3874652
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|c^2=ij,d^2=fhj,f^2=j,g^2=j,h^2=j,c^a=cg,c^b=ci,d^a=df,d^b=dfhj,d^c=dij,e^a=ehij,e^b=ei,e^c=eij,e^d=eh,f^a=fj,f^b=fj,g^a=gj,g^b=gj,g^d=gj,g^e=gj,h^b=hj,h^e=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287379
maximal normal subgroup of	#72287379

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