#72320973

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#486654
commutator, Frattini, cosocle	C₂×M₃₂	C₂⁴
prime radical	#7624864	C₂
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ #33435 ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

940 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:

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

Polycyclic presentation:

2^10|a^2=ghij,b^2=f,c^2=egi,d^2=hj,e^2=hij,g^2=ij,h^2=ij,i^2=j,b^a=be,c^a=ceghi,c^b=cefij,d^a=dgij,d^b=dhij,d^c=dghi,e^a=eh,e^b=eg,e^c=ej,e^d=ej,f^a=fghij,f^c=fghi,f^e=fj,g^a=gi,g^b=gij,g^c=gj,h^a=hi,h^b=hi,i^a=ij,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321055, #72321146, #72321336, …
maximal normal subgroup of	#72321055, #72321146, #72321336, …

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