#72225562

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#498562
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₂²×C₁₆	C₂⁴
prime radical	#513051	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ #12664 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

871 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fi,d^2=g,e^2=hij,f^2=j,h^2=ij,i^2=j,b^a=be,c^a=cg,c^b=ceghi,d^a=di,d^b=di,d^c=dg,e^a=eh,e^b=eh,e^c=ehj,f^b=fj,f^c=fj,g^a=gj,g^b=gj,h^a=hi,h^b=hi,h^c=hi,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225578, #72226849, #72226900
maximal normal subgroup of	#72225578, #72226849, #72226900

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