#72320274

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#376752
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ #10711 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1916 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ej,c^2=eij,d^2=ej,f^2=i,i^2=j,b^a=bg,c^a=cf,c^b=cfj,d^a=dh,d^b=dj,d^c=dij,e^c=ej,f^a=fi,f^b=fi,f^c=fi,f^d=fj,g^d=gj,h^b=hj,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	#72320350, #72320989, #72343946
maximal normal subgroup of	#72320350, #72320989, #72343946

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