#72225596

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	#136339
commutator	C₄×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₁₆	C₂³
prime radical	C₂×#508	D₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #12664 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

443 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=d,c^2=efgj,e^2=gi,f^2=hj,g^2=ij,h^2=j,i^2=j,b^a=bfgij,c^a=ce,c^b=cefhi,d^c=dj,e^a=eg,e^b=egj,f^b=fh,g^a=gi,g^b=gi,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225598, #72226897, #72226902
maximal normal subgroup of	#72225598, #72226897, #72226902

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