#72318677

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	#7623517	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2284 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)(5,18)(6,11)(7,12)(10,20)(13,22)(14,23)(15,27)(16,25)(17,26)(19,24)(21,28)(29,31)(30,32)(33,34)(35,41)(36,51)(37,43)(38,44)(39,45)(40,46)(42,53)(47,56)(48,60)(49,58)(50,59)(52,61)(54,62)(55,63)(57,64);(1,3,6,5,7,14,21,20)(2,8,11,18,12,23,28,10)(4,13,17,19,16,29,30,15)(9,22,26,24,25,31,32,27)(33,35,37,50,39,49,40,55)(34,41,43,59,45,58,46,63)(36,48,38,54,53,64,52,47)(42,57,61,56,51,60,44,62);(1,4)(2,9)(3,13)(5,19)(6,17)(7,16)(8,22)(10,27)(11,26)(12,25)(14,29)(15,20)(18,24)(21,30)(23,31)(28,32)(35,47)(36,52)(37,40)(38,53)(41,56)(42,44)(43,46)(48,55)(49,54)(50,64)(51,61)(57,59)(58,62)(60,63);(1,5,6,14,7,20,21,3)(2,10,11,8,12,18,28,23)(4,15,17,13,16,19,30,29)(9,24,26,31,25,27,32,22)(33,36,40,52,39,53,37,38)(34,42,46,44,45,51,43,61)(35,48,55,47,49,64,50,54)(41,57,63,62,58,60,59,56);(4,16)(9,25)(13,29)(15,19)(17,30)(22,31)(24,27)(26,32)(35,49)(41,58)(47,54)(48,64)(50,55)(56,62)(57,60)(59,63)

Polycyclic presentation:

2^10|b^2=f,d^2=fj,f^2=h,g^2=i,i^2=j,c^b=cg,d^a=dh,d^b=dhj,d^c=dhij,e^b=ej,e^c=ehj,f^c=fj,g^b=gi,g^c=gij,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	#72318707, #72318898, #72319381
maximal normal subgroup of	#72318707, #72318898, #72319381

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