#72312262

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	#10735
commutator	C₂⁴	C₄³
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#123452
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#568 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

3991 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)(4,18,20,45)(5,10,24,34)(6,11)(7,12)(9,21)(13,36,19,47)(14,27,16,29)(15,28)(17,30)(22,46,37,59)(23,33,44,56)(25,35)(26,38)(31,49)(32,50)(39,58,48,63)(40,51)(41,53,42,54)(43,55)(52,61)(57,60)(62,64);(1,3,6,15,7,17,25,43)(2,8,11,28,12,30,35,55)(4,19,22,48,20,13,37,39)(5,14,23,41,24,16,44,42)(9,26,32,52,31,51,57,64)(10,27,33,53,34,29,56,54)(18,36,46,58,45,47,59,63)(21,40,50,62,49,38,60,61);(1,4)(2,9)(3,13,17,19)(5,21)(6,22)(7,20)(8,26,30,51)(10,18)(11,32)(12,31)(14,38,16,40)(15,39,43,48)(23,50)(24,49)(25,37)(27,36,29,47)(28,52,55,64)(33,46)(34,45)(35,57)(41,61,42,62)(44,60)(53,58,54,63)(56,59)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=eg,f^2=j,b^a=bd,c^a=ce,c^b=cdij,d^c=dj,e^a=ei,g^a=gi,g^b=gj,h^c=hi,h^e=hj,h^g=hj,i^a=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312278, #72312895, #72313192
maximal normal subgroup of	#72312278, #72312895, #72313192

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