#72319889

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#377115
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13334	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10711 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2308 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)(3,14)(5,25)(7,28)(9,35)(10,32)(11,12)(13,43)(15,42)(16,50)(17,18)(19,26)(20,53)(21,56)(22,60)(23,61)(27,38)(29,45)(30,47)(31,49)(33,34)(39,63)(40,41)(46,52)(48,51)(54,55)(57,64)(58,59);(1,2,5,11,28,33,45,34,6,12,29,8,7,13,25,43)(3,15,14,44,30,42,26,35,18,49,17,37,19,31,47,9)(4,20,22,54,61,10,38,32,24,55,27,53,23,41,60,40)(16,51,50,62,59,48,56,39,52,64,46,36,21,57,58,63);(1,3)(2,9)(4,21)(5,26)(6,18)(7,30)(8,31)(10,39)(11,42)(12,35)(13,44)(14,45)(15,34)(16,23)(17,25)(19,28)(20,36)(22,50)(24,59)(27,46)(29,47)(32,57)(33,37)(38,58)(40,48)(41,63)(43,49)(51,54)(52,61)(53,64)(55,62)(56,60);(1,4)(2,10,12,41)(3,16,18,52)(5,27)(6,24)(7,23)(8,32,11,40)(9,36)(13,20,33,55)(14,46,17,50)(15,48)(19,21,30,59)(22,29)(25,38)(26,58,47,56)(28,61)(31,51)(34,54,43,53)(35,62)(37,39)(42,64)(44,63)(45,60)(49,57)

Polycyclic presentation:

2^10|b^2=e,d^2=efi,e^2=ij,f^2=ij,i^2=j,b^a=bf,c^a=cfh,c^b=cf,d^a=dg,d^b=di,d^c=di,e^a=ei,e^c=eij,e^d=ej,f^a=fi,f^b=fj,f^c=fi,f^d=fj,g^c=gj,h^d=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319909, #72320921, #72343980
maximal normal subgroup of	#72319909, #72320921, #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, 2, F.5);;
gap> SetPower(coll, 4, F.5*F.6*F.9);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
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);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
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.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);