#72307209

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	C₄	#21541
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#376255
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1881 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=egij,d^2=ehi,f^2=ij,g^2=i,i^2=j,b^a=bh,c^a=cf,c^b=cg,d^a=dij,d^b=dfgij,d^c=dfgi,e^d=ej,f^a=fi,f^b=fij,g^a=gi,g^b=gij,h^c=hj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72319909, #72320989
maximal normal subgroup of	#72307299, #72319909, #72320989

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