#72319284

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 ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2283 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)(3,13)(4,18)(5,21)(6,11)(7,12)(8,16)(9,15)(10,23)(14,25)(17,28)(19,26)(20,32)(22,30)(24,29)(27,31)(33,34)(35,41)(36,44)(37,43)(38,45)(39,46)(40,47)(42,56)(48,57)(49,59)(50,58)(51,60)(52,61)(53,62)(54,64)(55,63);(1,3,6,17,7,16,24,19)(2,8,11,26,12,13,29,28)(4,14,20,5,15,22,31,23)(9,25,27,21,18,30,32,10)(33,35,37,49,38,52,56,51,40,50,55,48,39,53,36,54)(34,41,43,59,45,61,42,60,47,58,63,57,46,62,44,64);(1,4)(2,9)(3,14)(5,17)(6,20)(7,15)(8,25)(10,28)(11,27)(12,18)(13,30)(16,22)(19,23)(21,26)(24,31)(29,32)(35,48)(36,37)(38,39)(41,57)(42,63)(43,44)(45,46)(49,50)(51,52)(53,54)(55,56)(58,59)(60,61)(62,64);(1,5,6,22,7,23,24,14)(2,10,11,25,12,21,29,30)(3,15,17,31,16,4,19,20)(8,9,26,27,13,18,28,32)(33,36,38,37,40,56,39,55)(34,42,45,63,47,44,46,43)(35,49,53,54,50,48,52,51)(41,57,62,60,58,59,61,64);(3,16)(4,15)(8,13)(9,18)(17,19)(20,31)(26,28)(27,32)(35,50)(41,58)(48,49)(51,54)(52,53)(57,59)(60,64)(61,62)

Polycyclic presentation:

2^10|b^2=f,d^2=fgij,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bhi,c^a=chi,c^b=cg,d^a=dhij,d^b=di,d^c=dh,e^b=ehij,e^c=ehi,f^c=fi,f^d=fj,g^b=gj,g^c=gij,g^d=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319316, #72319381, #72319579
maximal normal subgroup of	#72319316, #72319381, #72319579

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