#72307247

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#20451
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#354917
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2057 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=h,c^2=efi,d^2=ef,e^2=ij,f^2=ij,g^2=i,i^2=j,b^a=bg,c^a=ci,d^a=dij,d^b=dfgi,e^b=ei,e^c=ej,e^d=ej,f^a=fj,f^b=fi,f^c=fj,f^d=fj,g^a=gj,g^b=gi,g^c=gj,g^d=gj,h^a=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72319811, #72320817
maximal normal subgroup of	#72307299, #72319811, #72320817

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