#72319767

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#356610
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10711 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2314 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)(3,10)(4,11)(5,24,29,12)(6,13)(7,14)(8,16)(9,15)(17,44,26,36)(18,37)(19,38)(20,40)(21,39)(22,43,25,41)(23,42)(27,48,31,45)(28,52,35,46)(30,59,53,47)(32,49)(33,51)(34,50)(54,62,56,60)(55,64,58,61)(57,63);(1,3,6,18,32,4,7,23,9,21,34,57,8,20,33,19)(2,10,13,37,49,11,14,42,15,39,50,63,16,40,51,38)(5,25,27,55,53,17,28,54,29,22,31,58,30,26,35,56)(12,43,45,61,59,36,46,60,24,41,48,64,47,44,52,62);(1,4,8,21,9,20,32,3)(2,11,16,39,15,40,49,10)(5,26,30,25,29,17,53,22)(6,18,33,23,34,57,7,19)(12,44,47,43,24,36,59,41)(13,37,51,42,50,63,14,38)(27,58,35,56,31,55,28,54)(45,64,52,62,48,61,46,60);(1,5)(2,12)(3,17,21,26)(4,22,20,25)(6,31)(7,35)(8,30)(9,29)(10,36,39,44)(11,41,40,43)(13,48)(14,52)(15,24)(16,47)(18,56,57,54)(19,58,23,55)(27,34)(28,33)(32,53)(37,62,63,60)(38,64,42,61)(45,50)(46,51)(49,59)

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=efij,d^2=efj,e^2=ij,f^2=i,i^2=j,c^b=cf,d^a=dg,d^b=di,d^c=dij,e^c=ei,e^d=ej,f^b=fj,f^c=fi,f^d=fj,g^a=gj,h^d=hj,i^c=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319811, #72320989, #72343980
maximal normal subgroup of	#72319811, #72320989, #72343980

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