#72307706

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#346518
commutator, Frattini, cosocle	C₂×#131	C₂⁴
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

3967 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)(3,8,17,31)(4,9,20,34)(5,10,24,38)(6,11,21,35)(13,41,45,27)(14,47,49,28)(15,29,23,37)(16,30,25,39)(18,32,43,56)(19,33,48,59)(22,36,26,40)(42,55,54,64)(44,61,51,57)(46,62,52,58)(50,60,53,63);(3,13)(4,19)(5,23)(6,21)(8,27)(9,33)(10,37)(11,35)(14,16)(15,24)(17,45)(18,52)(20,48)(22,26)(25,49)(28,30)(29,38)(31,41)(32,62)(34,59)(36,40)(39,47)(43,46)(44,53)(50,51)(56,58)(57,63)(60,61);(3,14)(4,20)(5,24)(8,28)(9,34)(10,38)(13,16)(15,23)(17,49)(18,44)(19,48)(25,45)(27,30)(29,37)(31,47)(32,57)(33,59)(39,41)(43,51)(46,50)(52,53)(56,61)(58,60)(62,63);(1,3,15,50,21,43,24,16,7,17,23,53,6,18,5,25)(2,8,29,60,35,56,38,30,12,31,37,63,11,32,10,39)(4,13,42,14,48,52,22,51,20,45,54,49,19,46,26,44)(9,27,55,28,59,62,36,61,34,41,64,47,33,58,40,57)

Polycyclic presentation:

2^10|a^2=j,d^2=fghij,e^2=ij,f^2=i,i^2=j,b^a=bh,c^a=ch,d^a=dg,d^b=de,d^c=defij,e^a=eh,e^b=ei,e^c=ehj,f^b=fhi,f^c=fhj,f^d=fj,f^e=fj,g^b=gh,g^c=gh,g^e=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307752, #72307980
maximal normal subgroup of	#72307752, #72307980

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