#72310737

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#107114
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂³
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

2758 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)(4,17,20,37)(5,10)(6,12)(7,11)(9,19,32,24)(13,38,41,55)(14,27)(15,29)(16,28)(18,46,49,60)(21,33)(22,35)(23,34)(25,36)(26,40,30,44)(31,48,58,52)(39,61,47,64)(42,54)(43,56)(45,50)(51,59)(53,62,57,63);(1,3)(2,8)(4,13)(5,21,23,14)(6,16)(7,15)(9,30)(10,33,34,27)(11,29)(12,28)(17,38)(18,47,49,39)(19,44)(20,41)(22,51,50,42)(24,40)(25,43)(26,32)(31,53,58,57)(35,59,45,54)(36,56)(37,55)(46,64,60,61)(48,62,52,63);(1,4,5,18)(2,9,10,31)(3,13,14,39)(6,24,22,52)(7,20,23,49)(8,26,27,53)(11,32,34,58)(12,37,35,60)(15,41,21,47)(16,44,42,63)(17,45,46,36)(19,50,48,25)(28,55,54,64)(29,30,33,57)(38,59,61,56)(40,51,62,43)

Polycyclic presentation:

2^10|a^2=hj,b^2=g,c^2=d,f^2=j,c^a=cf,c^b=ce,d^b=di,e^a=eh,e^b=eh,e^c=ei,f^b=fhj,f^c=fj,f^e=fj,g^c=gh,g^d=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310741, #72310807
maximal normal subgroup of	#72310741, #72310807

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