#72306332

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	#146564
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

2765 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)(3,13)(4,19,21,56,24,57,23,38)(5,25,20,40)(6,12)(7,11)(8,16)(9,22,35,58,36,59,37,26)(10,27,39,29)(14,43,44,63,48,61,47,64)(15,49,17,53)(18,42)(28,41)(30,46,60,50,34,45,62,54)(31,55,32,51)(33,52);(1,3)(2,8)(4,14)(5,15)(6,16)(7,18)(9,34)(10,31)(11,33)(12,13)(17,20)(19,43)(21,44)(22,45)(23,47)(24,48)(25,53)(26,50)(27,51)(28,52)(29,55)(30,36)(32,39)(35,62)(37,60)(38,64)(40,49)(41,42)(46,59)(54,58)(56,63)(57,61);(1,4,20,58)(2,9,10,38)(3,14,15,50)(5,26,6,24)(7,21,27,59)(8,30,32,64)(11,35,40,19)(12,36,39,56)(13,34,31,63)(16,48,17,54)(18,44,55,45)(22,28,23,29)(25,57,41,37)(33,60,53,43)(42,62,49,61)(46,52,47,51)

Polycyclic presentation:

2^10|a^2=d,c^2=dfg,d^2=i,e^2=j,g^2=j,h^2=j,b^a=bij,c^a=ce,c^b=cf,d^c=dh,e^a=eh,e^c=ehj,e^d=eij,f^a=fj,f^c=fj,g^a=gh,g^b=gj,g^c=ghj,g^d=gij,h^a=hi,h^c=hi,h^e=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306338, #72306356, #72306381, …
maximal normal subgroup of	#72306338, #72306356, #72306381, …

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