#72319197

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	#7623520	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2265 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)(3,8)(4,9)(5,17)(6,10)(7,11)(12,20)(13,25)(14,21)(15,23)(16,24)(18,22)(19,28)(26,31)(27,30)(29,32)(33,34)(35,40)(36,44)(37,43)(38,41)(39,45)(42,53)(46,56)(47,55)(48,57)(49,58)(50,59)(51,60)(52,61)(54,62)(63,64);(1,3,6,14,7,5,13,19)(2,8,10,21,11,17,25,28)(4,12,15,26,16,18,27,29)(9,20,23,31,24,22,30,32)(33,35,37,49,38,48,39,54)(34,40,43,58,41,57,45,62)(36,47,52,63,53,46,51,50)(42,56,60,59,44,55,61,64);(1,4,6,15,7,16,13,27)(2,9,10,23,11,24,25,30)(3,12,14,26,5,18,19,29)(8,20,21,31,17,22,28,32)(34,41)(35,46,48,47)(36,51)(37,39)(40,55,57,56)(42,60)(44,61)(49,63,54,50)(52,53)(58,59,62,64);(1,5,6,19,7,3,13,14)(2,8,10,21,11,17,25,28)(4,12,15,26,16,18,27,29)(9,22,23,32,24,20,30,31)(33,36,37,52,38,53,39,51)(34,42,43,60,41,44,45,61)(35,47,49,63,48,46,54,50)(40,56,58,59,57,55,62,64);(3,5)(8,17)(12,18)(14,19)(20,22)(21,28)(26,29)(31,32)(35,48)(40,57)(46,47)(49,54)(50,63)(55,56)(58,62)(59,64)

Polycyclic presentation:

2^10|b^2=f,c^2=fi,d^2=f,f^2=h,g^2=i,i^2=j,c^a=cj,c^b=cg,d^a=dh,d^c=dhij,e^b=eh,e^d=ehj,f^c=fj,g^b=gi,g^c=gi,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319265, #72319316, #72319631
maximal normal subgroup of	#72319265, #72319316, #72319631

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