#72310758

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#201687
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#62852	C₂²
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=g,c^2=e,d^2=j,b^a=bg,c^a=cf,c^b=cfhj,d^a=dhj,d^b=dh,d^c=dj,e^a=ei,e^b=eij,f^b=fh,f^c=fi,f^d=fj,g^c=gh,g^e=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310767, #72310807, #72310871
maximal normal subgroup of	#72310767, #72310807, #72310871

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