#72303291

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10960
commutator	C₂²×C₄	C₄³
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#123452
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#3428 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

4039 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)(4,17,19,43)(5,21)(6,25,26,11)(7,12)(9,20)(10,24)(13,35,37,60)(14,33)(15,29)(16,41,40,30)(18,44,46,42)(22,28)(23,34,52,50)(27,38)(31,56)(32,48)(36,61,45,59)(39,55,51,58)(47,49)(53,64)(54,63)(57,62);(1,3)(2,8)(4,13,19,37)(5,22)(6,16)(7,15)(9,27,32,54)(10,33)(11,30)(12,29)(14,24)(17,35,43,60)(18,45,46,36)(20,38,48,63)(21,28)(23,51)(25,41)(26,40)(31,57,49,53)(34,58)(39,52)(42,61,44,59)(47,64,56,62)(50,55);(1,4,5,18,7,19,24,46)(2,9,10,31,12,32,21,49)(3,13,14,36,15,37,22,45)(6,20,23,47,26,48,52,56)(8,27,28,53,29,54,33,57)(11,17,34,42,25,43,50,44)(16,38,39,62,40,63,51,64)(30,35,55,59,41,60,58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303336, #72303452, #72303509
maximal normal subgroup of	#72303336, #72303452, #72303509

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