#72317876

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#146459
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂²×#124	C₂³
prime radical	C₂×#3500	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂²×#41 ◁ C₂×#3500 ◁ G
LECS	G ▷ C₂²×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2695 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)(4,19,20,55)(5,25,26,41)(6,28,29,12)(7,13)(9,21,36,57)(10,27)(11,30)(14,42,43,61)(15,47,16,49)(17,52,53,33)(18,34)(22,40,24,56)(23,38,58,37)(31,44,59,63)(32,48)(35,50,60,64)(39,51)(45,54,46,62);(1,3)(2,8)(4,14)(5,15)(6,17)(7,18)(9,35)(10,32)(11,39)(12,33)(13,34)(16,26)(19,54)(20,43)(21,44)(22,45)(23,50)(24,46)(25,47)(27,48)(28,52)(29,53)(30,51)(31,37)(36,60)(38,59)(40,42)(41,49)(55,62)(56,61)(57,63)(58,64);(1,4,7,22)(2,9)(3,14)(5,21,26,23)(6,20,29,24)(8,31,34,35)(10,19)(11,40)(12,36)(13,37)(15,44)(16,50)(17,43)(18,45)(25,55)(27,58,30,57)(28,38)(32,42,39,54)(33,59,52,60)(41,56)(46,53)(47,61,49,62)(48,64)(51,63)

Polycyclic presentation:

2^10|a^2=d,c^2=efg,e^2=j,g^2=j,h^2=j,b^a=bf,c^a=ce,c^b=cij,d^c=dh,e^a=eh,e^d=eij,f^c=fj,g^a=ghj,g^b=gj,g^c=gj,g^d=gij,h^a=hi,h^c=hj,h^d=hj,h^e=hj,h^g=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317908, #72318097, #72318162
maximal normal subgroup of	#72317908, #72318097, #72318162

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