#72307553

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346518
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3983 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:

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

Polycyclic presentation:

2^10|b^2=hj,c^2=f,d^2=h,e^2=i,f^2=ij,i^2=j,b^a=bhj,c^a=cg,c^b=ce,d^a=dh,d^c=defgh,e^a=eh,e^b=ei,e^d=eh,f^b=fi,g^b=gh,g^d=gh,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307601, #72307865
maximal normal subgroup of	#72307601, #72307865

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