#72307198

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#23848
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#359487
prime radical	#10585859	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5539 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)(4,18,22,45)(5,26,29,27)(7,32,20,47)(8,13,9,17)(10,37,14,23)(11,31,15,36)(12,34,16,35)(19,25,50,24)(21,51,28,49)(30,41,33,42)(38,46,43,40)(39,63,44,64)(48,54,52,56)(53,59,55,57)(58,62,60,61);(1,3,6,2)(4,19,22,50)(5,27)(7,32,20,47)(8,17,9,13)(10,38,14,43)(11,31)(12,34,16,35)(15,36)(18,24,45,25)(21,52)(23,46,37,40)(26,29)(28,48)(30,41)(33,42)(39,55)(44,53)(49,54)(51,56)(57,63)(58,61)(59,64)(60,62);(1,4,20,37,8,24,35,38,6,22,7,23,9,25,34,43)(2,10,12,18,17,46,47,50,3,14,16,45,13,40,32,19)(5,28,33,53,36,54,61,63,29,21,30,55,31,56,62,64)(11,39,41,52,26,57,58,49,15,44,42,48,27,59,60,51);(1,5,8,36,6,29,9,31)(2,11,17,26,3,15,13,27)(4,21,24,56,22,28,25,54)(7,33,34,61,20,30,35,62)(10,39,46,57,14,44,40,59)(12,42,47,60,16,41,32,58)(18,48,50,51,45,52,19,49)(23,55,43,64,37,53,38,63)

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=eghi,d^2=ij,g^2=ij,h^2=j,i^2=j,b^a=bf,c^a=cg,c^b=cghj,d^a=dij,d^b=dfhj,d^c=dfhi,e^d=ej,f^c=fj,f^d=fj,g^a=gi,g^b=gij,g^d=gj,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, #72319909, #72320639
maximal normal subgroup of	#72307301, #72319909, #72320639

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