#72307228

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#23573
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#359042
prime radical	#498872	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2959 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=eij,d^2=egij,e^2=j,f^2=ij,g^2=i,i^2=j,b^a=bf,c^b=ci,d^a=dg,d^b=dgh,d^c=dij,e^a=ej,e^d=ej,f^a=fi,f^b=fij,f^c=fj,f^d=fj,g^a=gij,g^b=gij,g^c=gj,g^d=gj,h^a=hj,h^b=hj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72320043, #72320921
maximal normal subgroup of	#72307301, #72320043, #72320921

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