#72287423

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623513	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,c^2=f,d^2=ghj,f^2=h,g^2=i,i^2=j,c^b=ch,d^a=dh,d^b=dh,d^c=dg,e^a=ej,e^b=eh,e^c=ei,e^d=eghj,f^d=fj,g^c=gi,g^e=gij,i^c=ij,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287479, #72287751, #72287811
maximal normal subgroup of	#72287479, #72287751, #72287811

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