#72303370

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#11043
commutator	C₂³×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#123862
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#3428 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

3807 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)(3,8)(4,18,22,46)(5,23)(6,26,27,11)(7,12)(9,21)(10,25)(13,37,19,47)(14,41)(15,31,16,30)(17,32)(20,49,51,45)(24,36,55,54)(28,39)(29,44)(33,40)(34,60)(35,52)(38,62,48,61)(42,57,43,58)(50,53)(56,64)(59,63);(1,3)(2,8)(4,19,22,13)(5,14)(6,16)(7,17)(9,33,35,28)(10,29)(11,31)(12,32)(15,27)(18,37,46,47)(20,48,51,38)(21,39,52,40)(23,41)(24,43)(25,44)(26,30)(34,59,53,56)(36,58)(42,55)(45,61,49,62)(50,63,60,64)(54,57);(1,4,5,20,7,22,25,51)(2,9,10,34,12,35,23,53)(3,13,14,38,17,19,44,48)(6,21,24,50,27,52,55,60)(8,28,29,56,32,33,41,59)(11,18,36,45,26,46,54,49)(15,39,42,63,16,40,43,64)(30,37,57,61,31,47,58,62)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=d,d^2=j,e^2=j,g^2=j,b^a=bf,c^a=ce,c^b=cegj,d^a=dj,e^a=ei,e^b=ej,f^c=fj,g^a=gi,h^c=hij,h^e=hj,h^g=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303384, #72303452, #72303514
maximal normal subgroup of	#72303384, #72303452, #72303514

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