#72282301

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#117591
commutator	C₂×#131	C₂²×C₄
Frattini, cosocle	C₂²×#131	C₂³
prime radical	C₂×#3584	C₂²
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=d,c^2=ef,f^2=ij,i^2=j,b^a=bg,c^a=cf,c^b=ce,d^c=dh,e^a=ej,e^b=eh,e^d=ej,f^a=fi,f^b=fh,f^d=fj,g^c=ghj,g^e=gj,g^f=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282419
maximal normal subgroup of	#72282419

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