Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #6453969 |
| commutator, Frattini, cosocle | C22×C8 | C25 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #60207 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
4771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,8)(4,17)(5,11)(6,12)(7,10)(9,20)(13,30)(14,27)(15,25)(16,26)(18,40)(19,41)(21,33)(22,31)(23,32)(24,37)(28,43)(29,45)(34,53)(35,55)(36,50)(38,51)(39,49)(42,54)(44,52)(46,56)(47,58)(48,61)(57,63)(59,62)(60,64);(4,18,20,43)(5,21)(6,23)(9,28,17,40)(11,33)(12,32)(13,34,37,58)(15,39)(16,38)(19,42,45,44)(22,46)(24,47,30,53)(25,49)(26,51)(29,52,41,54)(31,56)(35,57,61,59)(36,60)(48,62,55,63)(50,64);(4,19,20,45)(5,22)(9,29,17,41)(11,31)(13,35,37,61)(15,36)(18,42,43,44)(21,46)(24,48,30,55)(25,50)(28,52,40,54)(33,56)(34,57,58,59)(39,60)(47,62,53,63)(49,64);(1,3,7,14)(2,8,10,27)(4,13)(5,15,22,36)(6,16,23,38)(9,30)(11,25,31,50)(12,26,32,51)(17,24)(18,34)(19,35)(20,37)(21,39,46,60)(28,53)(29,55)(33,49,56,64)(40,47)(41,48)(42,57)(43,58)(44,59)(45,61)(52,63)(54,62);(1,4,5,18,6,19,21,44,7,20,22,43,23,45,46,42)(2,9,31,40,12,29,56,52,10,17,11,28,32,41,33,54)(3,13,36,58,16,35,60,57,14,37,15,34,38,61,39,59)(8,24,25,47,26,48,49,63,27,30,50,53,51,55,64,62)Polycyclic presentation:
2^10|b^2=h,c^2=h,d^2=hj,e^2=fgh,g^2=i,i^2=j,d^a=dh,e^a=eh,e^b=eg,e^c=ei,e^d=ef,f^a=fj,f^d=fj,f^e=fj,g^b=gi,g^c=gj,h^e=hj,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72249843, #72250830, #72281944, … |
| maximal normal subgroup of | #72249843, #72250830, #72281944, … |
| maximal quotient of | #72249954, #72250103, #72256866, … |
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, 2, F.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*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.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);