#72225604

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#134960
commutator	C₄×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂³
prime radical	C₂×#135	D₈
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#2170 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=d,c^2=egij,e^2=gij,f^2=hj,g^2=ij,h^2=j,i^2=j,b^a=bf,c^a=ch,c^b=ce,d^c=dj,e^a=ej,e^b=eg,f^a=fh,f^b=fh,g^b=gi,h^a=hj,h^b=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225606, #72226891, #72226896
maximal normal subgroup of	#72225606, #72226891, #72226896

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